Morse Homology (Progress in Mathematics 111)

Progress in Mathematics Volume 111 Series Editors J. Oesterlé A. Weinstein

Matthias Schwarz

Morse Homology

Birkhiiuser Verlag Basel • Boston • Berlin

Author: Matthias Schwarz ETH Zentrum Mathematik 8092 Zürich

Switzerland

Library of Congress Cataloging-in-Publication Data Schwarz, Matthias, 1967— Morse homology / Matthias Schwarz. p. cm. — (Progress in mathematics ; vol. 111) Includes bibliographical references and index. ISBN 0-8176-2904-1 1. Morse theory. I. Title. 2. Homology theory. Progress in mathematics (Boston, Mass.) ; vol. 111.

II. Series:

QA331.S43 1993 515'.73 — dc20

Deutsche Bibliothek Cataloging-in-Publication Data Schwarz, Matthias:

Morse homology! Matthias Schwarz. — Basel ; Boston ; Berlin : Birkhauser, 1993

(Progress in mathematics ; Vol. 111) ISBN 3-7643-2904-1 (Basel ...) ISBN 0-8176-2904-1 (Boston) NE: GT

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speci fically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 1993 Birkhduser Verlag Basel, P.O. Box 133, CH-4010 Basel, Switzerland Camera-ready copy prepared by the author Printed on acid-free paper produced of chlorine-free pulp Printed in Germany ISBN 3-7643-2904-1 ISBN 0-8176-2904-1 987654321

Contents List of symbols

viii

1 Introduction 1.1

1.2

Background

1

1.1.1

Classical Morse Theory

1

1.1.2

Relative Morse Theory

5

1.1.3

The Continuation Principle

7

Overview 1.2.1

The Construction of the Morse Homology

1.2.2

The Axiomatic Approach

8 8 10

1.3

Remarks on the Methods

14

1.4

Table of Contents

17

1.5

Acknowledgments

18

2 The Trajectory Spaces 2.1 The Construction of the Trajectory Spaces 2.2

2.3

21

Fredholm Theory

29

2.2.1

The Fredholm Operator on the Trivial Bundle

29

2.2.2

The Fredholm Operator on Non-Trivial Bundles

39

2.2.3

Generalization to Fredholm Maps

41

Transversality

41

2.3.1

The Regularity Conditions

42

2.3.2

The Regularity Results

49

CONTENTS

vi 2.4

2.5

Compactness

51

2.4.1

The Space of Unparametrized Trajectories

52

2.4.2

The Compactness Result for Unpaxametrized Trajectories 55

2.4.3

The Compactness Result for Homotopy Trajectories . .

63

2.4.4

The Compactness Result for A-Parametrized Trajectories

67

Gluing 2.5.1

Gluing for the Time-Independent Trajectory Spaces

68

2.5.2

Gluing of Trajectories of the Time-Dependent Gradient Flow

96

Gluing for A-Parametrized Trajectories

99

2.5.3

3

68

Orientation 3.1

3.2

Orientation and Gluing in the Trivial Case

104

3.1.1

The Determinant Bundle

104

3.1.2

Gluing and Orientation for Fredholm Operators

107

Coherent Orientation 3.2.1

Orientation and Gluing on the Manifold M

113 115

4 Morse Homology Theory 4.1

4.2

The Main Theorems of Morse Homology

133

4.1.1

Canonical Orientations

133

4.1.2

The Morse Complex

135

4.1.3

The Canonical Isomorphism

140

4.1.4

Topology and Coherent Orientation

153

The Eilenberg-Steenrod Axioms 4.2.1

4.3

163

Extension of Morse Functions and Induced Morse Functions on Vector Bundles

164

4.2.2

The Homology Functor and Homotopy Invariance

169

4.2.3

Relative Morse Homology

180

4.2.4

Summary

193

The Uniqueness Result

194

CONTENTS 5 Extensions

5.1 Morse Cohomology 5.2 Poincaré Duality 5.3 Products A Curve Spaces and Bartach Bundles A.1 The Manifold of Maps P1:121 (R, M) A.2 Banach Bundles on P1:12i (R,M)

199 200 202

207 214

B The Geometric Boundary Operator

221

Bibliography

229

Index

232

List of symbols

,C(X; Y) F(X; Y) /C(X; Y)

A eFA

Det

K H1,2(R , Rn)

CiTc Cr

Mx,y

110

Q90

1' • 7 #p re #p X

t

= -Yr

22 compactification of R to R U {±oo} space of continuous linear Banach space operators 29 space of Fredholm operators between Banach spaces 29 30 space of compact operators set of non-degenerate, conjugated self-adjoint opera29 tors on El' 29 matrix-valued curves with ends within S 35 set of special Fredholm operators equivalence class of operators from E tri, with fixed 35 ends determinant space of a Fredholm operator 105 surjectively extended Fredholm operator 109 22 Sobolev space of curves square-integrable together with the first derivative Banach space of sections in a vector bundle on R 24 with H 1,2 -quality 25 Banach space of sections with L2-quality space of smooth mappings with C"-compact-open 61 topology smooth mappings with compact support 41 R-smooth curves with fixed ends x, y 22 Banach manifold of H 1,2-curves with fixed ends 24 trajectory space associated to the time-independent 21 gradient flow of a Morse function f trajectory space associated to the time-dependent 21 gradient flow of a Morse homotopy A-parametrized trajectory space associated to a ho21 motopy of Morse homotopies Morse homotopy 21 unparametrized trajectory space associated to time54 independent gradient flow Banach space of variations of the Riemannian metric 45 75 differential DF„(0) of the Fredholm map Vf) loc in specific local coordinates on •101,2 x, y 52 time-shifting of a R-curve 68 gluing operation for trajectories 71 pre-gluing of trajectories, etc. 89 gluing operation for unparametrized trajectories 73 short notation for the triple of a broken trajectory and a gluing parameter

LIST OF SYMBOLS

A' a- [u, K]

ogeom[uxy ] A

F CA

O Critkf Ck(f)

Ck(f, fA) Hk (f, fA) D h*TM h*D 772 exp

O 77t .

frequently used cut-off functions 71 space of volume forms of a finite dimensional vector 104 space coherent orientation of the Fredholm operator K 129 along the curve u geometric orientation of a trajectory 222 set of equivalence classes of Fredholm operators along 129 curves transformation group for coherent orientations 129 set of coherent orientations 129 special natural isomorphism between orientations 127 set of critical points of a Morse function f with fixed 135 Morse index k k-th chain group generated by the elements of Critk f 135 k-th relative chain group associated to an admissible 184 pair of Morse functions k-th relative homology group of the relative Morse 184 complex associated to (f, fA) injectivity domain within the tangent bundle with 211 respect to the exponential map pull-back bundle with respect to the smooth curve h 211 pull-back of the injectivity domain with respect to h 211 fibre linearization of the exponential map with re- 215 spect to a Levi-Civita connection special bundle map on the tangent bundle associated 215 to the Levi-Civita connection covariant time-derivative of a vector field along a 215 curve

ix

Chapter 1

Introduction 1.1

Background

The subject of this book is Morse homology as a combination of relative Morse theory and Conley's continuation principle. The latter will be used as art instrument to express the homology encoded in a Morse complex associated to a fixed Morse function independent of this function. Originally, this type of Morse-theoretical tool was developed by Andreas Floer in order to find a proof of the famous Arnold conjecture, whereas classical Morse theory turned out to fail in the infinite-dimensional setting. In this framework, the homological variant of Morse theory is also known as Floer homology. This kind of homology theory is the central topic of this book. But first, it seems worthwhile to outline the standard Morse theory.

1.1.1

Classical Morse Theory

The fact that Morse theory can be formulated in a homological way is by no means a new idea. The reader is referred to the excellent survey paper by Raoul Bott [Bo]. The classical theory in its formulation by Marston Morse relates the analytic information by means of the critical set of the so-called Morse function to the global topology of the underlying manifold ( see [MI]). The local information about the critical points of a Morse function f E C c° (M, R) ( "Morse" means df ifi Om C T* M , i.e. at each x E Crit f the Hessian 112 f (x) is non-degenerate) is reflected by the Morse index, p(x) = dim Big - 112 f (x)

2

CHAPTER 1.

INTRODUCTION

as the number of negative eigenvalues of the Hessian operator at x E Crit f counted with multiplicity. This relation between analysis and topology is expressed in the form of the so-called strong Morse inequalities: We denote the Betti numbers and numbers of critical points of index k by

Ok =- dim Hk (M ,R) k = 0, . n = dim M , Ck # Critkf , Critkf = { x E Crit f 11(x) = k , k = O,. , n . Then the following holds:

ck - ck-i +

+ co cn - cn _i + + co

Ok Ok—i . . .

(30 for k = 0,

,n - 1 and

= X(M)

The key observation which finally provides this result is the change of the

homotopy type of the sublevel sets Ma ={xEM1f(x)a,} , when a crosses a critical value. For each critical point x with f (x) = a, a cell of dimension ii(x) is attached, when a crosses the value f (s) Ma+E Ma—E U

U e4(X) (c)

xECritfrlf —1

Starting from the absolute minimuml, one obtains a cell decomposition of M up to homotopy equivalence,

M

U e4(x)

xECrit f

The first steps toward an intrinsically defined Morse homology were taken successively by Thom, Smale and Milnor (40's-60's) by not only taking into account the number and types of cells in the cell decomposition indexed by the critical points but also the attaching maps. Let us henceforth stick to the convention of considering the negative gradient flow

//):118>(M-M (1.1)

— at 7P(t' x)

= -V f (,b(t,x)) ,

.) = idm

associated to a Morse function f on a compact, n-dimensional manifold M. Given this dynamical system, one is led to the stable and unstable manifolds

Wu(x) { p E M lim 0(t,p) = } , W3 (x) = {p E M lim V(t, p) = x } 1This exists if

M is assumed to be compact or f to be coercive, i.e. all M° compact.

3

1.1. BACKGROUND

associated to the isolated2 critical points x E Crit f. As early as the 1940's, R. Thom recognized that the disjoint decomposition of the compact manifold

M, xECritf

describes naturally the homotopically equivalent cell decomposition analysed in Morse theory. In fact, the unstable and stable manifolds are cells of dimension I.t(x) and n - i(x), respectively, with tangent spaces

Tx Wu(x) =

(x)

and

Tx 1473 (x) = Eig+H2 f(x) .

Unfortunately, this immediate cell decomposition does not represent directly a CW-complex, because in general the dynamical system might misbehave in such a way that the closure condition is not properly fulfilled Nevertheless, in the 50's, S. Smale found the crucial condition for this gradient dynamical system 3 , which allows us to describe all ingredients of a Morse theoretical homology, namely, the critical points as generators and the attaching maps of the associated Morse complex as boundary relations in terms of the unstable and stable manifolds. He discovered that the requirement of transversal intersections for all x , y E Crit f Wu(x) h 1473 (y) appears as a generic condition for the dynamical system. For instance, one can always find a generic Riemannian metric on M. This is the so-called MorseSwale condition. Given this condition, it is immediate that Wu (x)n W'(y) is a (11(x) - p,(y))-dimensional manifold which intersects the hypersurface f -1 (a) for any regular value a E (1(y), f (x)) transversally. Seeing that f -1 (a) carries a natural orientation by -V f , one is able to define a number n(x, y) for pairs (x, y) E Crit f x Crit f with relative Morse index it(x) - ii(y) = 1. n(x, y) is defined as the intersection number of Wu (x) and 1473 (y) restricted to the niveau surface f -1 (a). It is well-defined if we are provided with arbitrary orientations of the unstable manifolds Wu(x) , x E Crit f which imply orientations of the complementary manifolds Ws(x) given that M is oriented. In other terms, the short exact sequence

o Tp (wu(x) n Ws (y)) TpWu(x) (I) Tp W8 (y)

TpM 0 ,

associated to the transversal intersection PE Wu(x)

n ws(y) n f -1 (a) ,

yields the characteristic intersection number n(x , y). These numbers happen to describe exactly the attaching maps for the cells within the Morse complex. 2by the above assumption of non-degeneracy 3generalized to gradient-like vectorfields

4

CHAPTER 1. INTRODUCTION

This idea was taken up by J. Franks in [Fr]. In his paper he dealt with these attaching maps in terms of the Thom-Pontryagin construction. Thus one is able to define the following chain complex. The chain groups, graded by means of the Morse index, are defined as the free Abelian groups ck(f) = zeritkf , k =. 0, .. • ,n 7 generated by the critical points x E Critk f. Ck-i(f) is 'defined by Ck(f)

(1.2)

akx ==

E n(x, Y)Y ,

yECritk- i

The boundary operator ak

:

x E Critkf •

f

The key observation is that the associated homology Hk(C.(f), 8), k =0, ... ,n, recovers the standard homology of the underlying compact manifold. For a fruitful application of this interrelation, the reader is referred to the famous h-cobordism theorem, see [M2]. This isomorphism theorem for homology theories immediately implies the strong Morse inequalities. In fact, this is the essence of classical Morse theory, and the homological formulation is assigned to Thom, Smale and Milnor in equal parts. In 1982, E. Witten (see [W]) rediscovered this way of recovering the (co-)homology of the underlying manifold by solely taking into account the critical points of a Morse function, which are graded by their Morse index, and the isolated connecting orbits between those points of relative index 1, provided that the Morse-Smale condition holds. With this observation, Andreas Floer is said to have been led to the idea of considering Morse theory in thoroughly relative terms. Note that the above homology groups are constructed merely on the basis of non-degenerate critical points of a functional, which are graded by some Morse index, together with the information about the sets .A41,v of connecting orbits with relative index 1. Up to this moment the latter has been described in terms of the entire global gradient flow,

147'(x) rt) W 8 (y) . Also the proof of the fundamental isomorphism result, H(C.,(f), 0)`1'—' H(M) , relies on the analysis of the cell decomposition of the manifold based on the global gradient flow where M is assumed to be compact. In fact, it is largely due to Andreas Floer that the essential ingredients of this Morse homology can be extracted from the spaces of connecting orbits, the so-called trajectory spaces, in a completely relative way. Counting the isolated trajectories between the critical points x and y yields the intersection number n(x,y) within (1.1).

5

1.1. BACKGROUND

1.1.2

Relative Morse Theory

The prime motivation for the development of a purely relative concept of Morse theory is rooted in the existence problem of periodic solutions for a time-dependent Hamiltonian system considered under the non-degeneracy assumption. For a detailed presentation of this subject leading to the so-called Floer homology, the reader is referred to [Win [S], [S-Z1] and [S-Z2]. The variational approach consists of restating the existence problem of solutions of a differential equation as an existence problem of critical points of an appropriate functional. This functional ought to be related to the topology of the underlying spaces in a more obvious way. It is well known from the classical theory that Marston Morse applied it to the existence problem of closed geodesics as critical points of the energy functional

(7 : S l

M) 1-4 E(7) = f 17(01 2dt si

defined on an appropriate space of loops in the manifold M. Hence, one is naturally led to try to use these methods equally for the action functional A associated to the above Hamiltonian system,

A(7) = f rco —

1/(t, 7(t)) dt

D2 with respect to the symplectic form co and

5, : D2

M

7

'113D2 = : S 1

M .

One introduces an L2-scalar product on the vector spaces playing the role of tangent spaces to the loop space on which the functional A is defined. Then the zeroes of the gradient of A due to this scalar product can immediately be identified with the solutions of the Hamiltonian equation via

(1.3)

grad A (-y) = J(-'y — X Ho -y)

where

• XH

is the Hamiltonian vector field depending explicitly and periodically on the time t with period 1, and

• J is an almost complex structure associated to the Hamiltonian system with the symplectic structure w. The standard approach, however, which aims at the application of classical Morse theory in the infinite-dimensional setting, fails for the following reasons.

6

CHAPTER 1. INTRODUCTION

First, unlike the energy functional, the action is unbounded below and above4 , such that there is no absolute minimum with which to start a Morse complex in the sense of a cellular decomposition. Second, and even more striking, is the lack of a finite Morse index. The subspaces on which the Hessian operator is maximally positive and negative definite are both of infinite dimensions. Finally, the most natural candidate for a gradient of this action functional, grad A as mentioned above, does not even give rise to a well-defined flow on the loop space in question. This gradient behaves in a so-called unregularized way. Yet there is still hope for Morse theory. Namely, Andreas Floer recognized that in this situation all essential conditions for a Morse theory reduced to a purely relative version are fulfilled. This means a restriction to the relative gradient flow, i.e. a flow between fixed critical points x and y as limit sets, and to a relative Morse index which, roughly speaking, measures something like a codimension of the 'unstable manifold' of y with respect to the 'unstable manifold' of x. Here, we take the classical Morse theory considered above as a finite-dimensional model. This relative Morse index turns out to be finite. Setting up the nonlinear partial differential equation

(1.4)

u:Rx

lim u(s,.) = x(•) , y(-) respectively,

M,

u(s, t) + J&u(s,t) — JXH(t,u(s,t)) = 0 ,

Floer was able to handle finite-dimensional solution sets M which in Morse theoretical terms describe exactly the 'trajectory spaces' of connecting orbits between fixed critical points x and y. In fact, equation (1.4) might be restated formally using (1.3) as a -

as

u+gradAou = o.

Floer identified the local dimension of these trajectory spaces .A4 z ,y as the appropriate relative Morse index within this setting. The .A4 i,y prove to be manifolds provided that one is given a generic J. Using this relative grading on the set of critical points for the action functional and the sets of isolated connecting orbits5 , he found all ingredients for a homological concept known today as Floer homology.

4Note that the first term in the formula for A describes the area enclosed by y measured with sign. 5The local dimensions of the trajectory manifolds lead to such a well-defined relative grading only if certain conditions on the topology of the underlying manifold are fulfilled. Therefore, the Arnold conjecture still remains unproven in strict generality.

7

1.1. BACKGROUND

1.1.3

The Continuation Principle

The second fundamental part of the homology theory, besides the idea of relativity in Morse theory, is the continuation principle. This aspect has been investigated by Charles Conley in order to describe the topological type of invariant sets of flows in a way which is invariant under continuous deformations of the whole system. Concerning this topic the reader is referred to [C] and [C-Z]. The continuation principle appears as an essential idea within the theory of the Conley index. Due to this principle Floer was able to relate his homology theory based on the periodic solutions of a fixed Hamiltonian system to the topology of the underlying manifold. The idea proved very transparent. First, one has to choose a time-independent Morse function H° on the given compact manifold small enough in the C 2 -norm, so that there are no 1-periodic solutions of the Hamilton equation 'y(t) = JVH°(-y(t)), (note that XHo(t,p) = JVH°(t,p))

apart from the constant ones, namely the critical points of H°. Actually, the Morse homology associated to H° is naturally isomorphic to the Floer homology of H° by this identification (for a generic J). Conley's continuation principle is now integrated by the observation that any generic homotopy between H° and an arbitrary fixed Hamiltonian induces a unique isomorphism for the associated Floer homology groups in a canonical way, i.e. independent of the concrete choice of the homotopy. Additionally, one obtains the following functorial behaviour for these isomorphisms (Dp a : HF„(11,,) HF(H) between the homology groups for the Hamiltonians He, and Hp: (1.5)

po

= — 4)

and (locia = kIHF(H) •

Therefore, the family

H F.(He,) I He, an admissible Hamiltonian for Floer homology } together with these canonical isomorphisms forms what Conley named a connected simple system. By means of these isomorphisms one can carry out an inverse limit process and obtain the Floer homology HF (M) explicitly independent of the concrete Hamiltonian. This is the continuation principle which

provides homology groups intrinsically independent of a special Hamiltonian. Recalling that the time-independent Morse function H° considered above establishes the normal homology of M, provided that M is compact, it is now obvious that by Conley's continuation principle the Floer homology reproduces the ordinary homological information about M. Actually, this very inventive and fruitful theory by Andreas Floer not only concerns the Arnold conjecture on symplectic manifolds but was also

8

CHAPTER 1. INTRODUCTION

transferred successfully to 3-dimensional topology. Specifically this abstract method based on relative Morse theory and the continuation principle by Conley can be applied to the Chern-Simons functional in 3-dimensional gauge field theory thus leading to the instanton homology (see [F4 ] ), which is often denoted by Floer homology, also.

1.2

Overview

The main significance of this study, entitled 'Morse homology', is that it gives a presentation of a homological concept of Morse theory in strict analogy with Floer homology. Therefore it may serve as a finite-dimensional model. The results are based principally on the same methods and techniques and the prime emphasis lies on the detailed presentation of the underlying analysis. A survey of the analytic programme will be given in the next section on 'Remarks on the Methods'. Since the organization of this work is primarily governed by this analytic programme, the following overview is intended to convey an idea of the logical approach to the Morse homology rather than give a mere outline of the contents. The latter is postponed to the end of the introduction. Notwithstanding the first remark, the subject of this study is not only a finite-dimensional modelling of Floer homology. It also provides a construction of Morse homology as an axiomatic homology theory. This means an axiomatic theory in the sense of Eilenberg and Steenrod (see [E-S]). As a consequence of this axiomatic approach one immediately obtains the uniqueness result for Morse homology, that is a natural equivalence to any other axiomatic homology theory which is defined on the same suitably large category of topological spaces, like for instance singular theory. The advantage of this result based on the axiomatic approach is that, in contrast to the classical investigations by Thom, Smale and Milnor, it is valid also for non-compact manifolds and for all Morse functions simultaneously. The latter in particular is due to Conley's continuation principle.

1.2.1

The Construction of the Morse Homology

Before we present the actual axiomatic approach to Morse homology we shall first outline the development of the Morse homology groups themselves. The first step is to define the canonical chain complex associated to any fixed Morse function. This complex, which has been related to the strong Morse inequalities in the previous section, is called the Morse complex (7„(f) of f. In constrast to the treatment by Thom, Smale, Milnor and Witten, we shall prove the fundamental complex property, O2 =- 0, intrinsically. In other

L2. OVERVIEW

9

words, we shall deduce the identity

E

02x= E

(8z , y)(ey , x) z =

ti(z)=A(V)+111(11)=IL(1)± 1 directly from the analysis of the trajectory spaces for arbitrary smooth manifolds, without assuming orientability, and for general coefficients. The fact that Witten's idea of counting the isolated trajectories of the negative gradient flow (1.1) leads to a 0-operator of descending degree 1, relies on the R-symmetry of the trajectory spaces with respect to time-shifting. rye + r) yields a new trajectory with the same endpoints, provided that they are different. Hence, one obtains a set of isolated trajectories (modulo time-shifting) for relative Morse index 1. In the second step we shall introduce Conley's continuation principle. Motivated by the classical observation that the Morse complex for any Morse function on a compact manifold recovers the standard homology of the manifold, one is naturally inclined to search for an isomorphism between the graded homology groups associated to two different Morse functions. Moreover, similar to the canonical 8-operator, the isomorphism should arise from a given pair of Morse functions in a canonical way. In fact, this can be verified if we consider the isolated trajectories for the time-dependent gradient field (1.6)

= —Vht

0 ,

where f cx

f13

is

a (nearly arbitrary) homotopy between the two Morse functions. Since, due to explicit dependence on the time t, the solution curves of (1.6) are no longer endowed with time-shifting invariance, we exactly obtain isolated trajectories under generic conditions for critical endpoints ro, E Crit f' and ro E Crit ffl with relative Morse index Sa)

—

=0 .

In formal analogy with the definition of the 8-operator we can thus define the hornornorphism (1.7) V'xc, = n(x,„ x 0) xo

E

by counting the isolated connecting orbits in an appropriate way. This is the candidate for a morphism between the chain complexes C (fa) and C,,(ffi), which complies with the grading. Hence, there are two items to be analysed. First, one has to show that this homomorphism commutes with the 0-operators associated to f" and P. Second, one has to verify that the induced homomorphism (1)'?,' between the associated graded homology groups does not depend on the choice of the homotopy ht.

10

CHAPTER I.

INTRODUCTION

Quite interesting and revealing may be the fact that both aspects are treated by means of exactly the same techniques which are used for the proof of the (a' = 0)-relation. In fact, given two different homotopies and 14 between and fO, we choose a suitable homotopy

0

r

ii? 11"2 kJintroducing an additional parameter A E [0, 1] for the analysis. If we consider the trajectories of the associated )-parametrized and explicitly time-dependent gradient flow we shall obtain isolated trajectories for the relative Morse index —1. Actually, referring to the analogous definition of a homomorphism like in (1.2) and (1.7) we shall be led to a chain homotopy operator, from which we can conclude that both homotopies, and /4, induce the same morphism on the level of the homology groups.

q

Altogether, the induced morphism
{x} -, Ixs} g
(H

("\

Morse function on MI

gives rise to the definition of Morse homology groups independent of a fixed Morse function, (H. ( f a ) ) {f.} / „.4.2. . H.4f/f0rse(m) (1.8) We recall that that these homology groups are well-defined on arbitrary smooth manifolds provided that we consider Morse functions to be smooth functions f E C"(M,R) with non-degenerate critical points and compact sublevel sets

Ma = {x E M I f(x).a}, a el1 ; that is in short:

(1.9)

1.2.2

f E C cc (M, R) , df rh Om C TM and

f coercive .

The Axiomatic Approach

Let us now sketch a portrait of the functorial and axiomatic treatment of Morse homology. The fundamental idea behind this approach is based upon the concentration of the Morse homology by means of the continuation principle, thus providing homology groups which are explicitly independent of a certain

1.2. OVERVIEW

11

Morse function. By reverse arguing, this concentration, organized as a limit process, always allows us to choose specific Morse functions, which fit the given situation best, and to control any change of the function in a unique manner without deviating from a fixed equivalence class Rx1]. The following Morse functions, which are provided in a natural way on products and bundles with the local structure of a product, are of capital importance: If we are given manifolds (M , f) and (N , g) endowed with arbitrary Morse functions, then the operation (f ED g)(m, n) = (m) + g(n)

yields a Morse function on M x N and analogously in the case of a local product structure. The particular Morse function f identification

(1.10)

Critn (f

e g) = U

eg

obviously implies the

Critk f x Crit(n_ k)g .

04kti

Actually, this fundamental operation for Morse functions not only gives rise as a key feature to the entire functorial concept, it also forms the base for product operations within the (co-)homology theory which we wish to develop. The latter statement refers to the identity for the tensor chain complex

( f)

(g) = C. (f (13

g)

,

which is concluded from (1.10).

Relative Morse Homology and the Excision Axiom In order to develop an axiomatic homology theory we first have to begin with relative homology associated to a pair of manifolds, (M, A), where A forms as as any closed submanifold is equipped with an open tubular neighbourhood. We shall remain in analogy with the construction of the singular relative homology groups by means of suitable quotient complexes. Hence, we need an appropriate representation of subcomplexes in terms of a Morse complex. Essentially, this is provided by a restriction of the given Morse function to a subset of M, which Contains the critical points of the subcomplex in question and from which no trajectories of the negative gradient flow emanate. Hence, we take functions as so-called relative Morse functions on admissible pairs of manifolds, such that the gradient field restricted to aA forms an outward pointing normal field. From this we shall derive graded relative homology groups H, (M, A) together with the axiomatically required long exact homology sequence. Since for the definition of H. (M, A) we only need to know the Morse function of M on a neighbourhood of M\A together with the transversality of the gradient with respect to aA, the excision axiom is verified immediately according to the construction.

a submanifold of M Regarding the homotopy invariance restated below the homotopy axiom we may restrict our treatment to open submanifolds,

CHAPTER 1.

12

INTRODUCTION

The Functorial Property The functorial behaviour of the Morse homology, however, cannot be deduced directly from the definition. It is clear that arbitrary smooth maps between manifolds do not preserve the properties of a Morse function as specified in (1.9). Actually, it is at this stage that we have to refer to the facilities provided by the independence from the actual Morse function as mentioned above. In the first place, let us restrict ourselves to closed embeddings,

: M N As a consequence, we obtain a chain homomorphism (p„ between suitable Morse complexes in the following way. Let f be an arbitrary Morse function on M and let v be the normal bundle of the closed submanifold (p(M). Then the embedding cp and the Morse function qp(m) (x)

=

(x,x), p(m)

defined by means of any fiberwise Riemannian metric on the bundle v give rise to a Morse function

(1.11)

f (v(m), x) = 1(m) + qw (m)(x)

on the normal bundle v in the way that has been described above, and we obtain a canonical identification

Crit. (cp: f) = Crit.f . Now considering a tubular neighbourhood of cp (M), we are supplied with a function which can be extended to a Morse function on the entire manifold N, such that the Morse complex of f forms a subcomplex of C4.((p. f) up to identification. Hence, every closed embedding induces a chain homomorphism for Morse homology. Referring to similar investigations by means of the product property

(1.10) this functorial concept may be extended to projections Pr2 : M x N N Then, finally, the homology functor for arbitrary smooth maps ça M N in general is obtained from the factorization

(1.12)

p= (id , (p) o pr2

into the closed embedding of the graph manifold and the projection.

13

1.2. OVERVIEW

The Homotopy Axiom and the Dimension Axiom Finally, we finish this outline of the axiomatic approach with some remarks on the remaining two axioms. Regarding the homotopy axiom, this is immediately deduced from the continuation principle. Actually, the entire construction of the Morse homology is based upon the invariance under homotopy via the identification isomorphisms 44a . Thus it is sufficient to consider homotopies of closed embeddings which give rise to homotopies of Morse functions. Perhaps more novel and interesting is the investigation of the dimension axiom. Let us first review the framework for the Morse complex as it was analysed by Smale and Witten. They assumed the underlying manifold to be compact and orientable. As a consequence they obtained a complex both for the Morse function f and for —f, that is to say that they were able to consider the ascending and the descending boundary operator on equal terms. In fact, based upon these assumptions for M one naturally recovers the symmetry known as Poincaré duality. This symmetry is abolished immediately if one generalizes for non-compact manifolds or coefficients in Z in the non-orientable case. We observe that in general the right behaviour as homology theory, i.e. concerning the descending boundary operator for the negative gradient flow, is related to the coercivity of f as stated in (1.9). Obviously, any continuous function on a compact manifold is coercive. In summary it can be stated that coercivity is the crucial feature which underlies the dimension axiom, that is to say, Hrorse (m,

z) z

if M is pathwise connected. Products and Further Extensions As we have observed above, the natural operation f (0 g for Morse functions in some local framework plays a crucial role within this kind of Morse homology theory. It is thus quite revealing to see that this feature gives rise to a rich variety of extensions to further items of algebraic topology, for example to product operations. Making use of the well-known dualization concept by means of the hom-functor, which provides a cohomology theory starting from chain complexes, together with the corresponding functorial behaviour with respect to the diagonal map : M M

X

M

we recover the construction of the cup-product. At the end of this chapter we shall give a brief outlook on the possibilities of the treatment of features like the Thom class or the Euler class

14

CHAPTER 1.

INTRODUCTION

associated to a vector bundle within the framework of the Morse (co-)homology theory. This possibility is due to the concept of relative Morse functions. If we are given any vector bundle ir :E--■ M endowed with a Riemannian metric, the latter gives rise to the fiberwise quadratic, decreasing function —qm (x) --- —(x, x), , 7(x) = m E M .

Combining this function with any arbitrary Morse function f on the base M we obtain a relative Morse function f e (—q) for the pair (E, E\{01). Then the natural identifiCation Critkf 'L-2 Critn+kf ED ( -0 should yield the starting point from which one can develop further elements of algebraic topology concerned with vector bundles.

1.3

Remarks on the Methods

As we have already mentioned, the essence of this treatise on Morse homology is the methodical relation to Floer homology. The main techniques which will appear within the analysis refer to Banach calculus on Banach spaces and Banach manifolds, for example the fundamental contraction mapping principle, as well as to Fredholm theory, which means either the theory of linear Fredholm operators or of nonlinear Fredholm maps. The framework for this calculus is typically infinite-dimensional. The purpose is to give a largely self-contained presentation. We also add a detailed deduction for the infinite-dimensional Banach manifolds which arise during the discussion and for which it is particularly troublesome to give an appropriate reference.

Before we come to a listing of the analytic programme which ought to serve as Ariadne's thread through the labyrinth of infinite-dimensional analysis, we first give an outline of the prime investigations. The main interest of this approach to Morse homology in contrast to earlier treatments is the analysis of the trajectory spaces of negative gradient flow. Instead of analysing such spaces geometrically as transversal intersection manifolds Wu (x) rh W 8 (y), we shall treat them as zero sets of a nonlinear operator

(1.13)

F=

a

on a Banach manifold Pxl 'y2 of curves. This happens in strict analogy to Floer's manifolds of instantons. In our case Px1 '112 denotes the Hilbert manifold of H I '2-

curves6 y: R U {±oc} -- ■ M 6These are square integrable together with the first weak derivative.

1.3. REMARKS ON THE METHODS

15

with fixed endpoints

7( --00) = X,

Y(+00) = y •

It forms the infinite-dimensional fundamental space for our analysis. Now, exactly like in the case of the nonlinear Cauchy-Riemann operator with the additional Hamiltonian term in Floer's approach, the linearization of the elliptic operator F turns out to be of Fredholm type. This is largely due to the asymptotic ellipticity provided by the non-degeneracy of the Hessians at the critical endpoints. The crucial observation which gives rise to relative Morse theory is the statement that the local index of F is given by the relative Morse index p(x) — tt(y). This follows from the analysis of the spectral flow of the linearization of V'f as the curve between the Hessians H2 Ax) and H2 f (y) with values in the set of endomorphisms in the pull-back bundle of TM along the trajectory -y. The next step is to identify the classical Morse-Smale condition as of the zero section as a value for the non-linear operator F. Thus, provided a generic Riemannian metric, we can conclude the manifold property of the trajectory space .A.4‘., y = F -1 (0) from the implicit function theorem. The convincing advantage of this Banach calculus is expressed as the fact that the analogous results on the trajectory spaces for the time-dependent gradient flow follow directly from a simple and immediate generalization of F to

the regularity

a We will recognize that these analytical methods allow us to unify the proofs of • the chain complex identity 02 = 0 • the chain morphism identity aova =vaaa • and the chain homotopy identity 040" — iff3"aa = (DP — 4,ga

to a large extent.

The intrinsic deduction of these three fundamental identities

is a consequence of the study of the respective trajectory spaces. That study will involve the questions of "compactness" "gluing" and "orientation" for the totality of the trajectory spaces

{

(x, y) E Crit f x Grit f } .

We shall obtain the above three identities from discussions of coherent orientations on all these trajectory spaces and of cobordism relations with respect to these orientations. In fact "compactness" and "gluing" are complementary concepts which yield the result that the trajectory manifolds without boundary' 7with respect

to

the H1,2-topology

16

CHAPTER 1.

INTRODUCTION

Jvt.1 y are compact up to the splitting-up of trajectories, that is, up to the existence of sequences which converge to so-called broken trajectories in the Cipo°,topology. These are tuples (u 1 , , u,,) of trajectories from lower-dimensional trajectory spaces with coinciding critical endpoints, ,v— 1

= us,1_1(—cx)) for i = 1, and u i (—co) = x, ti,(+co) =- y .

"Gluing" amounts to exactly the reverse process, namely to mapping such broken trajectories ecjuipped with an additional suitable parametrization into the appropriate higher-dimensional trajectory space, ,A4 r , y x

My,z D

(U, V)

u#p V E

J'vtx,z,

P Po ,

such that the sequences (u#,,„v) nE N with pn —4 co correspond exactly to the aforementioned obstructions to compactness. This complementarity of gluing and compactness engenders a cobordism relation between the trajectory manifolds in the sense of the Ci -topology. Especially for isolated trajectories, compactness guarantees that there are only finitely many trajectories to count. Finally, 'coherent orientation' describes an orientation concept for these trajectory spaces which is respected by those cobordisms. This concept is based upon analysis of the so-called determinant bundle, which is a line bundle on the topological space of Fredholm operators for given fixed Hilbert spaces. The idea is to generalize the classical notion of the orientation of a finitedimensional manifold within this special framework of Fredholm maps. Since the tangent spaces of the trajectory manifolds may be identified with the kernels of the linearized Fredholm sections F, this new concept also yields an orientation for an appropriate composite of kernel and cokernel of a non-surjective Fredholm operator. Thus, orientations of trajectory spaces are well-defined even if no regularity (i.e. the property of being a manifold) is guaranteed. This turns out to be exactly the right concept for an oriented cobordism analysis. We thus obtain the proofs for the above three fundamental identities by means of a calculus of oriented cobordisms for the trajectory spaces associated to the negative gradient flow of a Morse function. Moreover, it is worth mentioning that the class of Fredholm operators which after trivialization are of the type

a A : H1,2(R , Rn) L2(R , Rn) 3i+ A(±00) has real non-zero eigenvalues ,

forms the key feature for the entire methodological foundation of this approach to Morse homology. Discussions of the analogous analysis of the Floer type instanton spaces, which correspond to the trajectory spaces, can be found in [F1], [F2],

1.4. TABLE OF CONTENTS

17

[F3], [F5], [F6], [Mcll], [S], [S-Z1] and [S-Z2]. Whereas in these treatments only Floer homology groups with coefficients in Z2 are considered, a new joint paper [F-H], by Floer and Hofer, presents the method of the so-called coherent orientation of certain Fredholm operators, which allows one to generalize the theory for coefficients in Z. After these remarks on the methods we now present the fundamental analytic programme which is exactly the same as for Floer homology: 1. analytical setup, definition of the trajectory spaces 2. analysis of the index problem

3. transversality: regularity results 4. compactness 5. gluing 6. coherent orientation.

1.4

Table of Contents

As a guide to the reader, we once again briefly sum up the contents of the monograph: Chapter 2 comprises the entire analytical work on the trajectory spaces except for the orientation problem. The organization is firmly related to the above analytic programme. The purpose is to develop the complementarity concept of compactness and gluing as described above. Section 2.1 contains the construction of the trajectory spaces as zero sets of appropriate sections in Banach bundles on Banach manifolds of curves. Section 2.2 analyses the Fredholm property of these sections and identifies the Fredholm index as the relative Morse index by means of the spectral flow. Section 2.3 is devoted to the transversality problem. The regularity result is obtained from a parameter version of the theorem of Sard-Smale. Section 2.4 gives an account of the effect of splitting up of trajectories as an obstruction to the strong compactness of the trajectory spaces. This gives the first half of the crucial complementarity of compactness and gluing. Section 2.5 establishes gluing as the reverse operation by means of the Banach contraction mapping principle and finally yields the first basic results on the cobordism relations among the trajectory spaces. Chapter ,3 on the subject of orientation, is separated from Chapter 2 on trajectory spaces because of its methodical complexity. In this chapter we shall develop an orientation concept for Fredholm operators which generalizes the notion of the orientation of a finite-dimensional manifold, so that together

18

CHAPTER 1. INTRODUCTION

with the outcome of Chapter 2 we obtain a theory of oriented cobordisrns for our trajectory manifolds. Chapter 4 finally contains the main proofs for the construction of Morse homology. Section 4.1 is devoted to the three fundamental identities stated above, which are proved by means of the oriented cobordisms calculus. In Section 4.2 we go through the axioms of Eilenberg and Steenrod and finally come to the uniqueness result in Section 4.3, where the link to any other standard homology theory is established. Chapter 5 presents an account of the possibilities for recovering the product operations and the naturally present Poincaré duality. Finally, in this chapter we give an outlook on possible further developments toward algebraic topology on vector bundles. Chapter A of the appendix gives all the technical details which are necessary to construct and handle the Banach manifold of curves P1'1,2 together with some Banach bundles on Pxl , which turn up within the analysis of trajectory spaces. Chapter B, finally, is intended to yield the missing link between the geometrical definition of the boundary operator according to Thom, Smale, Milnor and Witten and the definition within this present work. There we verify the equivalence of the coherent orientation concept with the classical concept of transversal intersection of the unstable and stable manifolds for the gradient flow Wu(x) rti W 8 (y), as far as the trajectory manifolds are concerned.

1.5

Acknowledgments

The author was partially supported by DAAD-Procope and the Graduiertenkolleg `Geometrie und Mathematische Physik' at the Ruhr-Universitat Bochum. This monograph is an outgrowth of my diploma thesis completed in January 1992 under the direction of Prof. H. Hofer at the Ruhr-Universitht Bochum. I am very much indebted to my teachers Helmut Hofer and Ralph Stocker, from whom I learned much of this material and whose stimulating courses have generated great enthusiasm for the subject. I would like to express my deep gratitude to Professors Hofer and E. Zehnder for convincing me to publish this work and for their constant encouragement and invaluable support. I am also very grateful to Martin Reimann at the University of Bern for his warm hospitality and his stimulating interest. I profited from innumerable discussions with friends and colleagues. I am particularly indebted to Dietmar Salamon, Graeme Segal, Jiirgen Jost and Norbert A'Campo for their advice and encouragement. Many people generously devoted considerable time and effort to reading the manuscript and helping me improve the exposition. Among

1.5. ACKNOWLEDGMENTS

19

them I am especially grateful to Stefan Ha.ndzsuj, Claudia Putz, Prof. Gordon Wassermann, Tilmann Wurzbacher and to Birkhduser Verlag. Zürich, April 1993 Matthias Schwarz

Chapter 2

The Trajectory Spaces 2.1

The Construction of the Trajectory Spaces

In this chapter we shall construct the trajectory manifolds needed for the definition of Morse homology. These trajectory spaces associated to the negative gradient flow correspond to the moduli spaces and instanton manifolds in Floer homology theory. In the most general form appearing in the following analysis, these trajectory spaces are composed of the solutions of the non-autonomous ordinary differential equation

(2.1)

V ht = V1+ Ii1t i 2 1Vht i 2

The scaling factor in the denominator has been chosen in order to guarantee compactness in the case of time dependency (see the section on 'compactness').

To give a preview, the following trajectory spaces will be investigated: • The time-independent case ht ==_ f yields the negative gradient flow of f. It serves to define the 0-operator in the Witten complex. The time-independent trajectory spaces are denoted by • The time-dependent case of a Morse homotopy (see Definition 2.40) h" 14

with = 0 for Iti ?.. R will be studied in order to construct the transformation homomorphisms (1) 0' which provide the means for identifying homology groups associated to different Morse functions. Analogous

22

CHAPTER 2. THE TRAJECTORY SPACES to the time-independent case, we denote these time-dependent trajectory spaces by Msh no

• Finally, we shall analyze the parametrized case of the so-called A-parametrized trajectory spaces used to define the chain homotopy operator, i.e.

Mryo

(Al

7)

[0, 1] xRxM

where HO

, A

-Y E

E [0, 1 1 } 5

R is a generic homotopy of Morse

homotopies. By close analogy to Floer homology we shall employ methods for the analysis of the required properties which basically rely on the calculus of Fredholm operators and their orientation.

In order to obtain the results about the trajectory spaces in an analytical way they will be equipped with a local Banach space structure. In fact, we will represent them as sub-manifolds of suitable Ba,nach manifolds. On the one hand, with respect to the Fredholm operators mentioned above, it seems worthwhile to study a local H 1,2 (R, Rn)-structure. On the other hand, we are interested in compact -curves, that is to say in mappings with

lim -y(t) = x and resp. y E M . t.±00

Therefore, it is with respect to endpoints x, y E M chosen in a specific way that we shall construct the Banach manifolds P1 ,1/2 with H 1,2 (R,Rn)-structure on which the entire calculus is founded. It should be emphasized that due to this differentiable structure these manifolds of maps cannot be identified with subspaces of the Hilbert manifold H 1 ' 2 ({0, 1], M) of which the reader familiar with Klingenberg's variational approach to the geodesic problem might be reminded.

Definition 2.1 We compactify R asIli = RUI±ool equipped with the structure

of a bounded manifold by the requirement that h: {

Ftt

-> [- 1, 1] , t

be a diffeomorphism.

Additionally, given arbitrary endpoints x,y E M we define the set of smooth, compact curves CZ as

= CZ

M) = fu

u(+oo) = x ,u(+co) = y .

Owing to this differentiable structure on R we obtain the following characterization of the asymptotic decrease of Cc°(-1k)-smooth functions:

2.1.

THE CONSTRUCTION OF THE TRAJECTORY SPACES

23

Lemma 2.2 For each function f E C l (N,R) there is a constant c(f) > 0, such that the following estimate holds:

c(f) 3

Ifi(t)1

for all t E R .

(1+ t 2 )

Proof. Due to the definition we have f 0 h-1 E

C'([ -1,1],R). Hence, we are

able to choose c(f) to be c(f) = sup l(f o h-1 ) 1 1 < [-1,11 and conclude the proof with ((h -1 )' 0 11) (t) =

1

(1+ t2 )i .

h'(t)

Actually, the boundedness of f and f' already yields the following: Corollary 2.3 Given A E Cl(rt,GL(n, )), there is a constant c(A) > 0 such that the estimate

11A 8 111,2

c(A)

1 8 111,2

holds for all s E H 1,2 (R,Rn), where (As)(t) = A(t) • s(t), t E R. Corollary 2.4 Let f E C 1 (R, ) satisfy the asymptotic condition f(±oo) = O. Then it holds f E H1,2(R ,

Proof. It is clear from Lemma 2.2 that f' E L2 (111, R) holds. The computation for arbitrary to, t E R relying on Lemma 2.2 ft

f (t) — f

(to)

=

t (1+7 c(f)_2) cl-rr

f ( T),T 1

c(f)

L

=

t-2

I

implies the estimate If(t)1

c(f)t -2 , t E R

due to the assumption that f(±co) = 0 as to

±oo.

L2 (R, R) . 1We

may even derive

f

E

H1- 1 as

becomes clear in the proof.

This proves f E

24

CHAPTER

2. THE TRAJECTORY SPACES

e E Vec(R) be a Cc° 01TO-smooth, finite-dimensional _ vector bundle on R and let 0 : e R x Rn be a smooth trivialization.2 Then using the induced one-to-one mapping 0. between the associated vector spaces

Definition 2.5 Now let

of sections we are able to define

H111 ,2(e

)

0471 (H1,2(R,R a)) _

et, 1 (s)1 s E fl l '2 (R,Rn )

.

It is immediate from Corollary 2.3 that 0, induces a Banach space topology on the vector space Hile (e) in a way which is independent of the particular choice of the trivialization 0, because the change of trivialization from 0 to .7-5under the assumption of smoothness is represented by some A = o E

C'(, G L (n, R)) . Once again, it seems worthwhile to underline that we have not deduced this space 1/R1 '2 W c Hilo'c2 (0 from a finite measure on ift but from toplinear isomorphisms with H 1 '2 (R, Rn). The discussion of the properties of this so-called section functor H11 '2 , which are fundamental with respect to the development of the Banach manifold we desire, shall be deferred to the section on manifolds of maps in the appendix. At this stage, we shall be satisfied with a brief definition of this Banach manifold P1'112 as we claimed it to be. Starting off with the exponential map on the complete Riemannian manifold M, exp :TM DD

M,

where D stands for an open and convex neighbourhood of the zero section, we denote by h*D the induced open and convex neighbourhood of the zero section in the pull-back bundle h*TM, for any smooth, compact curve h E Cm (R,M). Definition 2.6 Regarding the the map

(2.2)

Sobolev embedding Hilc;c2 C° we are led to

(1, M) exp h 14 ' 2 (VD) s exp os, (exp os)(t) = exp h (t ) .8(0

which is well-defined for all h E Ca) (R,M). Thus, we are able to define ps1 :id2(R , Af)

= { exp os E C° (R, M) I s E Hllit '2 (ha D), h E CZ9 (R, M)

.

The following proposition will be the first result of the discussion in Appendix A concerning the structure of the set Pl:v2:

2R is contractible.

2.1. THE CONSTRUCTION OF THE TRAJECTORY SPACES

25

Proposition 2.7 The set of curves 1)11 c C ,M) with given endpoints x,y E M is equipped with a Banach manifold structure via the atlas of charts

exph

hECzv (li,M) '

Additionally, the following inclusion relations hold:

(2.3)

der px1 :y2(R , m.)

C°( i,

dese cx13,y

Moreover, there is a countable sub-atlas.

We shall obtain the following representation of the tangent space,3 (2.4)

Tpx1 :y2

HR1,2 (pl:y2* Tm )

U H2 (s*TM) sEP!:;,

This is a Banach bundle on 7)1 ,y2 with H 1,2 (R, Rn) as characteristic fiber. By strict analogy to Hilz ' 2 we are able to define a section functor (as defined in the

appendix)

vecc oo() Ban which is endowed with a Banach space topology given by L 2 (R,Rn). This lends itself to the Banach bundle (2.5)

Llt (Pg* TM) = U Li(s*TM) . .E .Pg

Then, the second fundamental proposition developed in the appendix is Proposition 2.8 Let f E C°° (M,R) be an arbitrary, smooth real function on M. Then, given critical points x,y E Crit f as endpoints, the gradient field V f induces a smooth section in the L 2 -Banach bundle,

La MI :v2 *Tm)

F

os

1-4

Actually, the study of the trajectory manifold MI y is founded upon exactly this section F. If we substitute the time-independent vector field Vf by the time-dependent, suitably scaled vector field Xt 3The extension of appendix.

=

H/R1'2 and LR2 to

Vht

the set of curves

PI:v2 will also be studied

in the

26

CHAPTER 2.

THE TRAJECTORY SPACES

the associated, more general section Fx, yields the manifold of time-dependent trajectories in an analogous manner. It should be emphasized that by the definition of ht (see Morse homotopy) X t is again a gradient field for Iti > R. In this most general situation throughout the whole treatment we obtain the local representation of the section F

(2.6)

Floc : 11511 '2 (-y*D)

Floc

La(-y*TM), -y E

(ow-- V(t) + g (t, e(t))

from the local trivializations of the bundles HR1 ' 2 (1)1.*TM) and La ("Pl'* TM). Here, g : R x ry*D -y*TM is smooth and fibre respecting at each t together with g(±oo, 0) = O. Moreover, the asymptotical fibre derivatives F2g(±oo, 0) are conjugated to the Hessians of f° and ff3 at xa and xo, respectively. In the following section we will concentrate on this type of map Fia„ between Banach spaces. There, it will turn out that it is merely the asymptotic behaviour at the ends of the trajectories which proves significant. The fundamental link with the trajectory spaces of the (negative) gradient flow is now provided by the following simple proposition. Given critical points x and y of f E Cc°(M, ), it holds that

L2 (Pa.1 '1,2*TM) are exProposition 2.9 The zeroes of the section F : actly the smooth curves which solve the ordinary differential equation =

s

and which submit to the convergence condition

lim s(t) = x, lim s(t) = y . t—co Proof. Due to the assumption, each zero of this section represents a weakly differentiable and continuous curve with 7= —V f o y -

as its weak derivative. The asserted property follows by means of iterative insertion and differentiation from the smoothness of f. In principle, this is nothing more than elliptic regularity. In fact, compared to the analogous but less trivial result in Floer homology, this principle should be mentioned. For the converse, due to the definition of "Px1 ' and Lemma 2.2, it is sufficient to show that for any solution s of the differential equation obeying the convergence condition demanded above, the composition

."§-=soh-1:[-1,1]—M

2.1. THE CONSTRUCTION OF THE TRAJECTORY SPACES

27

is continuously differentiable. S" satisfies o h) (t) = S(t) (1 + t2 )i .

(2.7)

Thus, we have accomplished the proof if we have shown t lim S(t) • (1+ t 2 )i = O. This property of the asymptotic behaviour of the gradient trajectories is immediate from

Lemma 2.10 Let X : U(0) — ■ IR.' be a continuously differentiable vector field defined on a neighbourhood of 0 E JR and let 0 be a critical point of X such that the linearization DX(0) is non-degenerate and symmetric. Then, there is an E > 0 such that every solution è = X (s)

(2.8)

with

lim s(t) = 0

t,co

satisfies the following estimate: There are constants c> 0 and to c IR depending on s and satisfying is(t)I ce't for all t to • (2.9) Proof of the lemma. We start by defining the non-negative function

a(t) = ( s(t), s(t)> , that is by means of (2.8):

a' (t) = (X (s(t)), s(t)). One more differentiation

yields a" (t) = (D X (s(t)) • S(t), s(t)) + (2( (s(t)), X (s(t))) .

(2.10)

The continuous differentiability of X at 0 gives rise to the identity (2.11) IX(s(t)) — DX(0) • 8(01

R(s(t)) • Is(t)1

with Jim R(s) = 0 x—o

and thus to the following estimate: (2.8),(2.10)

a" (t)

(DX (s(t)) • X (s(t)), s(t)) (DX (0) • [X (s(t)) — DX (0)s(t) D X (0)s(t)] , s(t))

+ K[D X (s(t)) — D X (0)] • X (s(t)), s(t)) (2.11) (DX(0);(t), DX(0)S(t))

11DX( 0)11 R(S(t)) WO, S(t))

- (51X (S(t))1 1.9(01 for t t 0 (8), so that s(t)

ci"(t)

D X (0) (t) 1 2

ro is small enough. That is

6 is(t)I 2 for arbitrary t

to(e), E > O .

CHAPTER 2.

28

THE TRAJECTORY SPACES

Given any arbitrary 5 > 0, the step before the last stems from the continuity of DX(s(t)) and the symmetry of DX(0). Applying (2.11) once again and choosing a suitable ô = (E) leads to the last estimate. Due to the fact that DX(0) E GL(n,R), we have

c= inf 1DX(0) • xl > 0 . xesn-1 If we now choose

(2.12)

we obtain the crucial inequality

a" (t)

Is(t)1 2 = cc(t)

for all

t

to(c,$)

Hence, the assertion is concluded from the following maximum principle: Setting ao (t) = a(to ) and t°) A(t) = a(t) — ao(t) we reproduce the estimate (2.12) in the form

A"(t)

c A(t) for all t „>- to

with A(to ) 0 and tlim. A(t) = O. According to these properties the function A cannot have a positive local maximum at any To > to. This leads to the estimate

a(t).<, a(to) e - `1(t-t0) for all t

to(c, s) 4

concluding the proof. Actually, because of the recursive relation given by S = X(s) the gradient trajectories turn out to have an exponential asymptotic decrease in each derivative. Therefore, they are --differentiable infinitely many times and Proposition 2.9 has been proven. This proposition also holds immediately for the time-dependent trajectories, because for large times we have to deal with ordinary gradient fields. After these preparations, we have managed to represent the trajectory spaces as zero sets of smooth sections in Banach bundles on Banach manifolds. In order to extract the fundamental result about .A/11,y , namely the manifold property, we need a further transversality condition:

F: Px9

L2 (P*TM)

has to intersect transversally the zero section in the L2-bundle. This so-called Morse-Smale condition is not fullfilled in general. But if we admit minor variations of the Riemannian metric on M, this regularity condition appears to be generic with respect to the parameter set of Riemannian metrics on M. This result shall be investigated in the section on transversality. 4This lower bound to depends on s solely via the condition I s(t)I < ro, for t ro(X).

and given a small

to(s)

2.2. FREDHOLM THEORY

2.2

29

Fredholm Theory

The aim of this section is to develop the Fredholm properties of the operators which are needed in order to define the trajectory manifolds M ° and .A.4x//::3y ,. At first, we shall be content with considering linear operators on the trivial vector bundle R x Rn, of the type

(FAs)(t) = b(t)+ A(t) • s(t) , with s E Hi' 2 (R,Rn) and A E

2.2.1

CP, (R,End(Rn)) .

The Fredholm Operator on the Trivial Bundle

Definition 2.11 An operator A E End(Rn) is called conjugated self-adjoint

if it is self-adjoint with respect to some scalar product. Additionally, we shall henceforth use the following notations: H1,2 (R , Rn) ,

y L2 (R , wz) ,

S = {A E GL(n,R) I A is conjugated self-adjoint} , I AI = A(±oo) e S } A = {A G C° ( -11,End(Rn)) Here, S describes the set of the Hessians appearing in Morse theory. Thus, the following definition makes sense: Let p(a) # (a(A)

n -)

for A E S

be the so-called Morse index of A, where cr(A) denotes the finite spectrum of A. Moreover, we regard A as normed vector space with respect to We now wish to study the following map:

(R, End(Rn )) 3 A 1--+ (FA : X -- ■ Y) , (FA s)(t) = h(t) + A(t) • s(t) . Here, F:

(18, End(Rn)) --- ■ .C(X; Y)

describes an affine map, which is continuous due to the estimate (FA — F8)(s)II 0 =

ilA

Moo 11 3 110

(f 1(A — B)s1 2 dt) irA

hence,

— FBIL(x ,r)

Il A BlLc

,

CHAPTER 2.

30

THE TRAJECTORY SPACES

Actually, the central statement of this section is that F associates a Fredholm operator5 FA E .F(X;Y) to each A E A, so that the following formula holds:

ind FA

= p(A) —1i(A) .

This is the so-called relative Morse index, which is determined by the spectral flow of A. Proposition 2.12 Given A E A, the linear operator FA : X —4 Y Fredholm operator.

is a

The essential ingredient of the proof of this Fredholm property is provided by the following fundamental lemma about semi-Fredholm operators, bounded linear operators with closed range and kernel both of finite dimensions.

Lemma 2.13 Let X, Y and Z be Banach spaces and F E L(X;Y), K Z) and c> 0 with

E

MX;

(2.13)

), for all x E X .

c(

Then F is a semi-Fredholm operator. Although this lemma is a well-known, standard technique for investigating Fredholm operators, we nevertheless give a proof for the reader who is not quite familiar with this type of analysis. Proof. First, let us regard an arbitrary sequence (xk)kEN within the set

S

E kerFl lixil x , 1) .

Then assumption (2.13) and the identity

Fxk = 0 for all k E N yield the estimate

—xilix

I , for all k,l E N c IIK(x k — M IHz

Owing to the compactness of K, the sequence (Kxk)kEN and therefore also the sequence (xk)kEN have convergent subsequences. Thus, S is compact and the kernel of F has finite dimension. Second, due to this finite dimension the theorem of Hahn-Banach implies the existence of a closed sub-space X0 c X satisfying the relation

ker F etop X0 -= X . 5This is a bounded linear operator with both kernel and cokernel having finite dimensions.

2.2. FREDHOLM THEORY

31

Let us now start with a sequence (Fxk)kEN C R(F) converging in Y; that is to say, without loss of generality, we can assume y E Y.

(xk)keN C Xo, Fxk

If (xk)keN is unbounded, we switch to the normed sequence il x

til . Thus, it

holds without loss of generality that (xk)k,EN, IXktx — 1, liFxkl

Y k

for all k E N.

According to assumption (2.13) there is a convergent subsequence of (xk) kEN because of the compactness of K and the convergence of (Fxk)kEN . Upon consideration of a further subsequence, this leads to = 1 and Fx =- 0

x, jIx

in contradiction to the construction of Xo. Therefore, the sequence (xk)kEN must be bounded, so that the same argument, based on (2.13), again yields a convergent subsequence. This implies the identity

y

Fx

for some x E X0 .

Hence, R(F) is closed in Y. Proof of proposition 2.12: The proof consists of four steps. We shall obtain the semi-Fredholm property from the asymptotic behaviour via a compact operator as in Lemma 2.13 by expressing the action of the operator in question during a compact interval of time. After this main part of the proof, the finiteness of the dimension of the cokernel follows from a formal adjunction of FA. Step I: Let A E A be a constant map from Ft to S. Then, there is a constant c > 0, such that the estimate c liFAsHo

holds for all s E X .

Proof. By .F : Y -1-% Y we denote the Fourier isometry, where we use the notations X = 111,2(IR , d'72 kL. ) and Y = L 2 (R,Cn) throughout this calculation. In particular, it holds that f(h)(t) = i t.F(s)(t), t E R Additionally, let w : F(X) that we obtain the identity

Y be the operator w(s)(t) = t • s(t), t E R so

.F(FA(s)) and therefore (2.14)

for all s E X .

(iw + A) o F(s)

FA=1F-1 o(iw+A)o..F: X-0/.

CHAPTER 2. THE TRAJECTORY SPACES

32

Now let us assume A E S, so that A0 = min la-(A)I > 0 holds. We further consider B 2(w ) : Cn

Cn B(W)•X=(1+)iX,CJ E

11Z,

B2 (w) = iw+ A, (4.) E R , and hence

- 0 g (B2(w)) = i + a(A) . As a consequence, the inverse B2(w) -1 exists and satisfies

1 sup III32-1 (w)11 = sup j A-1 =AEo(A)-Fiu., IA1 xca(B2(w))

1 VAF, + cv 2

Summing up we obtain

1+ \/ A 02 + (412

11B1 (w) B2-1 (w) I

1 max (- , 1) = c(A) . Ao

We henceforth conceive B1 , B2-1 E (R,End(Cn)) as multiplication operators in £(X; Y), so that we obtain the inequality

(2.15)

11 13 1

I I L—

c (A )

Putting all this together we are led to the required estimate by the combination of

11s1 1 =

+ w2 F sI1 02

=

11F-1131 BV1F ..F-1 B 2 F 4 02

with

(F-I BIBV.Fe,

.Fri) o

=

II11B 1 B

lieu0 11 711lo •

Due to (2.15) this yields

11•F-1B1B1F IL(x,y)

c(11 )

such that we conclude the estimate

IslI

c(A)11F -1B2 Fsllo

and owing to (2.14) the inequality which we asserted above.

Step 2: Given an arbitrary A E A, we find constants T> 0, c(T) > 0, such that it holds that

IISIII

c(T) VAsilo

for all s E X, s1[_T, T] = 0 .

2.2. FREDHOLM THEORY

33

Proof. Step 1 provides us with the estimates

Ilsili < c(A) IIFA± sil o

for all s E X .

We therefore define c = max (c(A+) c(A - )) . Given an arbitrary € > 0 and A E A, there is a T, > 0 large enough such that

IIAA(t)II < e for all t -T and e for all t T . iiA+ - A(t)ii

<

Restricting ourselves to 8± E X with 91[T,œ) = 0 and s+I(,,T1 = 0, respectively, we obtain the inequalities

FA± 81 110

11FAS ±

110

il(FA±

FA)

SI

II0

I FAS ± 110

•

Let us now consider s E X equipped with the property sk_T,71 = O. Then we find s± as above fulfilling s -= s+ + s- such that

Ms1 + 1sIIi C 11 FA - S - 110 1 FA+S+ 110 11FAS10 cc (li s- 1l0 ± OS + 110 IIFAS+ 110 C 11FAS II0 CE 1 8 110

1141

C

This at last implies

1 8111

C IIFAS110

cf

together with an appropriate T(e)

and for an e

lIsII

-

CE

11FA 5 ij0 , for all s E X with sii_T(,),T(,)] = 0 .

Step 3: Given any A E A, there is a Banach space Z and a K E 1C(X; Z), c> 0 satisfying iixiix IIK xil z) for all x E X . IIFAlly

In fact, FA is a semi-Predholin operator. Proof. Let T(A) be as provided in Step 2. Then it holds that

f

+ A s1 2 dt

- T

f (1.§I 2 + 2(§, A s) + IA .91 2 dt -T

Y (1

— I A S1 2

dt

- T

due to the computation of

IAl 2 + 2 (§, A s) + 1A S1 2 ?, 2

- 124,9 1 2

•

CHAPTER 2. THE TRAJECTORY SPACES

34 from

1.4+ 2A81 2 ?.••

11,4(t)11 • 1.8(01

Therefore by means of 1A(t) 8 (t )1 maxHT,T1 11A(t)11 , we conclude

1.4 + A 812 di

f

.

f 1.§1 2 dt -

—T

af

—T

and setting

'sr at

a

=

.

—T

Hence, there is a c > 0 satisfying

(2.16)

(1s1 2 + (à1 2 dt —T

—T

Defining a cut-off function 0 E C

1 0, fi(t)

cf (181 2 + 1S + As1 2 dt .

= 1 1,

(R, [0, 1]) with the proprties

tI „>- T+1 and 0(t) 0 for It' E (T,T + 1)

we are able to combine the estimate (2.16) with Step 2:

Ps+ (1 - 0) 8 11i

110 8 111 + 11( 1 - fi)s111 11FA ((1 - 0) 8 )110) c (110 8 110 + 11FA(0 8 )110

11.9 111

for a c> 0 large enough. That is

Is 1 1 1

c

(110311 0 +2

I )911 0

+

( 11811L2([_T-1,T+1])

110FA 8 110 ± 1 ( ( 1 - 0)FAslio)

11FA 8 11o) -

Considering the following composition of a continuous restriction map and the Rellich compact embedding, rest

K: H1 '2 (R,1Rn)

Cpt.

11 1,2 ([-T- 1, T + 1],Rn)

L2 ([-T- 1,T +1],Rn) = Z ,

we obtain

K X Z Si [—T-1,T+11

E L2 ([—T

—

1,T

+ 1 ])

as a compact operator. Thus, Lemma 2.13 provides the asserted semi-Fredholm property.

2.2. FREDHOLM THEORY

35

Step 4: FA is a Fredholm operator. Proof. Up to this point, we know that FA has a finite-dimensional kernel and a range that is closed in Y. Hence, the cokernel of FA is a Banach space satisfying the relation

coker FA""=" R(FA) i

,

because Y is a Hilbert space. Thus, let r E R(FA) -L , that is (7- , + A s)0 = 0, for all s

H 1.2

In particular, we can deduce

(r , 0)0 = —(A t r , 0)0, for all 0 E

(R, Er) .

But by definition, this means the weak differentiability of r. Hence, r E L2 is once weakly differentiable with r = At. r e L2 . Summing up, we obtain

r e 1471 '2 with

r = At • r .

Consequently, we know that r e X and r E ker F_Ai, because —At E A. Therefore, there is a toplinear isomorphism

cokerFA

ker F_ A, .

By strict analogy with Step 3 we conclude dim ker F_At < Do thus proving the proposition. D Having seen that the operator FA is a Fredholm operator, the following analysis will result in a proposition expressing the associated Fredholm index in ternis of the relative Morse index. At this point it should be metioned that only the spectral flow of A e A appears to be significant. This means the 'number of eigenvalues which change the sign'. In the proof we shall transform A successively without altering the Fredholm index into an operator of a shape that can be analysed easily. Concerning the focal class of operators we shall henceforth use the following notation: Definition 2.14 Prom Proposition 2.12 we consider the subset

E = F(A) = {FA E f(X;Y) IA E A} c .F(X; Y)

and denote the equivalence class of operators from E with respect to the relation 13 1 = Al by = A±} , A C A. eFA {FB E I

The following lemma will play a decisive role for the index theorem.

CHAPTER 2. THE TRAJECTORY SPACES

36 Lemma 2.15 Given any F subspace of (X; Y).

E

E, the class e F is contractible within E as a

Proof Let F = FA° E E be arbitrary and define 0 = 9 . Then we study the map

xe

:

0 with ti ( r, FA)

= FA(r)

that is (1 — r) • A + T • FA(T ) E 0 for all T E [0, 1], as A(r) ± = A ± At, . A(r)

We obviously

have K(O, •)

Thus, we

Ide

and ic(1, .) =

FA, .

must show the continuity of n in both variables, so, let us start with

(2.17)

1imr=

r

n•—■ oo

and

nlimôo FA n

within 0 ,

= FA

that is

lim 11FA„ — We

ilc(x ; Y) =

must verify lim liFA( r ) FAn(rn)11L(x,y)

n—too

Assuming

an

c>0

and a subsequence

FA( T ) —

we find

a sequence

Pink -= 1

=o •

(nk)kEN satisfying

>c

FA(T.k>11c(x,Y)

for all k E N

(unk)kEN C X, such that

and

I (AN — An, (Tr, )) • unk IL

Owing to assumption 2.17 we

e for all k

E

N

are led to convergence by

110

(A(T) — An k (Tnk )) • //ilk = [( 1 — 7") A + TA° — (1 — rnk) • Ank k = j (A Ank ) ' Un k + ( 7- — Tn. k) AO (T„ A — TA + Tnk A nk — Tnk A) • unk lio

A01 • Unkil0

-

—

kil

b

rn

+ IT T1 11.1 11 11 11C (X X) + irn ki —+

0, for k

11A0I 1 r(X ;Y)

I1F

A — FAnk

G(X ;Y)

00

in contradiction to the assumption of the above c.

0

2.2. FREDHOLM THEORY

37

Z turns out As a conclusion from this lemma, the index map ind : E to be constant when restricted to the class OFA . In other words, ind(FA) is determined uniquely by the endpoints A± E S. With this in mind we now come to the final step of the calculation of the Fredholm index. Actually, we carry out an equivalence transformation, namely an appropriate conjugation of the operator FA: Let A E A, that is, A± E S. Then there exists a C± E GL(n, R) satisfying

C± A ±(C±) -1 = diag() , • • An+ ) so that the eigenvalues

4 E C(A) sgn Ap

are ordered by sign, i.e. 3

sgn AP+ , .

Additionally, the ends CI can be chosen in order to satisfy det C± > 0, so that both C+ and C - lie in the same pathwise connected component of GL(n, R). Thus, there is a curve

C G Cœ (R, GL(n,R)) with ends C(±oo) = C, which additionally is asymptotically constant. This means C+, t T c _ t _T for some T > 0 .

C(t)

We shall henceforth denote by C both multiplication operators

Cx:X- ■ X and Cy:Y--*Y, C•s . It is obvious that C represents a toplinear isomorphism and that the following identities hold:

(CyFAC;r1 )(s)(t) = C(t) (A + A(t)) • (C-1 (t) • s(t)) = 40+ (C(t) A (C')(t) + C(t)A(t)C -1 (t)) • s(t) (t) •

(FcA (c-i Here, Si (C-1 )(t) = 0 holds for II

T . Thus we compute the ends as

(cg(c-1)+cAc--1)± =

diag

, AZ) = D I

and we obtain the relation CyFAC x-1 E eDi •

Owing to the fact that Cx : X =+ X and Cy : Y =-* Y are isomorphisms, the identity ind (CyFAC;(1 ) = ind FA

CHAPTER 2. THE TRAJECTORY SPACES

38

follows from the composition rule of Fredholm indices. So, to sum up, it is sufficient to compute ind FD for an element D E A having ends of DI = diag (4, ). This enables us to reduce the proof of the following proposition to a nearly explicit computation. Proposition 2.16 Given any A E relative Morse index,

A, the Fredholm index of FA equals the

ind FA =- 1t(A) — p(A) . Proof. It is sufficient to compute ker FA and coker FA when A is of the shape

A(t) = diag (Ai (t), sgn

(+co)

,

(t)) where

sgn A, 4_1(±oo), i = 1,... ,n —1,

and À(t) = const for It„>.. 1 . Sn (t)) lies Under these circumstances s E H 1,2 (R,Rn), S(t) = (Si (t), within the kernel of FA, s E ker FA, exactly if it represents a global solution of the system of differential equations Mt) = —A, (t) Mt), i with the bounded functions A, : R times shows that =- —A,(t)

1,...,n ,

R. Now, explicit calculation for large

t E R and 0

s, c B-1,2(R , R)

holds if and only if p

/V 0, as

-A- t

si (t) = { - - ; t '

t<1

e ., , t > 1

Having in mind the ordering of the eigenvalues by sign, we compute dim ker FA

= #

E { 1 , • • • • n} À

< O und A tl > 0 1

= max (A(A — ) —,u(A+),0)

.

By the same method of computing we obtain the formula dim coker FA = max (tt( — A — ) —

0) = max (p(A + ) —

0)

ker F_At = ker F_A. Computing the differfrom the isomorphism coker FA ence between the dimensions of the kernel and the cokernel amounts to proving the asserted formula ind FA = 11 (A - ) — 1 1 (A +)• It seems worthwhile to point out that by this method of calculating the index, we have executed transformations in order to simplify the operator

2.2. FREDHOLM THEORY

39

FA. These transformations preserve the Fredholm index but do not leave the dimensions of the kernel and the cokernel invariant in general. Regarding the question of such invariance transformations we refer to Section B in the appendix, where we shall make use of another method to obtain a diagonal shape comparable to Proposition 2.16. In fact, that technique will preserve dim ker FA and dim coker FA, too, and we obtain the formulas dim ker FA dim coker FA

2.2.2

= # fic E

{1, , n} Pt; < 0 and A 14,- > 0}

= # cE 1, ,n} jA k— > 0 and )t i- < 01 . {

The Fredholm Operator on Non-Trivial Bundles

We now discuss the general case of smooth vector bundles on 11 endowed with a Riemannian metric, so that H 1,2 () and L2 ( ) are well-defined. The problem is that we have to generalize the time-derivation by a covariant derivation which is a priori noncanonical and which might disturb the Fredholm property. Definition 2.17 Suppose that we are given the Riernannian bundle R with a covariant derivation V induced by a connection. Then we denote by Ecv the

class of operators FA H1,2()

L2()

FAs = VtS-F A s ,

with A E C) (End(0), such that A(±oo) is non-degenerate and conjugated self-adjoint on The first question is, whether the Fredholm property is well-defined for all FA E Ecv. Choosing any trivialization of on Ft, we obtain the following representation for the covariant derivative: (2.18)

Vlrivs = j s + rs

c ,

s

H

1,2 (R7R n)

where r E C'(t, End(R")) stems from the Christoffel symbol associated to V and the respective trivialization. It is now decisive how VIr iv transforms under a change of the trivialization 4) E C cn° (R, Gl(n)). We compute (2.19)

cr i vIriy( os )

(

a +0

-

1 (i,, + 0 -1 r0) s

Therefore, the trivialized operator FA% transforms as (2.20)

-1 .p.tAr:;, 4) _ gt 0 -1 ())+ 0 -1 r0

+ 0 -1"

Since due to the k-differentiability ci) vanishes asymptotically, i.e. ci)(±co) = 0 according to emma 2.2, the only problematic issue which might afflict the Fredholm property is the asymptotic behaviour of r, i.e. f(±co).

CHAPTER 2. THE TRAJECTORY SPACES

40

Definition 2.18 We call a covariant derivation V on a R-bundle e Fredholm admissible if any trivialization gives rise to asymptotically vanishing r-terms, i.e. F(±co) = 0. As we observed above, this property holds for every trivializations if it holds for one. Lemma 2.19 Let e = u* E be a pull-back bundle with respect to a 11-curve u E M), where 7r : E M is a smooth Riemannian bundle on the manifold M with a fixed covariant derivation V. The induced covariant derivation u*V on e is Fredholm admissible. Proof. For the sake of simplicity we also denote the curve in local coordinates by u. With respect to these local coordinates on M together with a local triviafization of E the covariant derivation V on E is represented by means of the Christoffel symbol r, i.e.

V„w(p) = Dw(p) • y + F(p)(y, w) . This yields the following local representation for the induced covariant derivation u*V, where we denote u*V ,di by Vt

(2.21)

Vte(t) = (t) + F(u(t)) (ù(t), e(t)) .

For these identities concerning the covariant derivation the reader is also referred to [KN. Since the above F-term is now given by

r(t) =_ r(u(t)) (Jim, • ) , we immediately deduce the identity F(±oo) = 0 from ù(±œ) = 0. Hence, u*V is Fredholm admissible. Thus, provided an admissible V, the Fredholm property of the above defined operator FA E Ecv depends solely on the asymptotic behaviour of A, i.e. A(+oo). We obtain this property for FA via an arbitrary triviafization

: 11Z x

e, that is q'FAq E eF„ C Etriv

,

B± = Here,

induces the toplinear isomorphisms

X

H1 ' 2 (0, (Oxs)(t) = 0(0 •

Oy : Y -2, L2 (0

analogously ,

which are again denoted by 0. Finally, we are able to transfer all results from the trivial case in a manner which is independent of the particular trivialization: Proposition 2.20 Given a Fredholm admissible covariant derivation V on e, every FA E Ecv is a Fredholm operator with the index

ind FA = P(11— ) — p(A+)

41

2.3. TRANS VERSALITY

2.2.3

Generalization to Fredholm maps

Finally we are able to state the result which we need for the class of Banach space maps upon which the entire analysis is founded.

Definition 2.21 Let e be a Riemannian C°° (R)-vector bundle with a Fredholm

admissible covariant derivation as above and let 0 C the following:

be open and satisfying

There is a 5E Cr ( ) with s(R) c 0, and Ot 0 and convex for all t E

n et

is open

that is smooth and fibre respecting, Further, let us assume a map f 0 0) = 0 and the fibre derivatives are non-degenerate such that it holds f(±, and conjugated self-adjoint, i.e. ti(D2 f (Ioo, 0)) is well-defined. Considering such maps f we are able to define Pred(e, V) =

: H1 '2 (0)

L 2 (e)

(Fs)(t) = ts(t) + f (t,s(t)) , f : 0 — ■ e as above

Additionally, let us denote by H1'2 (0) = {s E H 1 ' 2 ( subset of 11 1 ' 2 (e).

)I

C 01 the open

Proposition 2.22 Each F E Fred(, V) is a Fredholm map with its differential DF(so) E for all so E so that the local Fredholm index is determined by the formula ind DF(so) = p(D2 f (— , 0)) — it(Dz.f (-Foo, 0)) . Proof. It is clear from the properties assumed for f 0 identity holds for all so E H 1 ' 2 (0):

DF(S0) = FD2 foso

that the following

E ECV •

Hence the proof is immediate from Proposition 2.20.

2.3 Transversality In this section we shall obtain the result that the trajectory spaces come equipped with the structure of finite-dimensional manifolds. As it has been already mentioned above, we obtain this property only if we restrict to a generic set of metrics or homotopies of Morse homotopies, which in this situation operate as parameters. As regards this genericity result the fundamental technique for the proof is based on a parametrized version of the theorem of Sard-Smale, which is adapted to the given situation of sections in Banach bundles.

CHAPTER 2.

42

2.3.1

THE TRAJECTORY SPACES

The Regularity Conditions

Definition 2.23 We call a subset E c X of a Baire space g-set, if it is a countable intersection of open and dense subsets and therefore dense, also. We then call a set G C X generic with respect to a condition of the points of X if this condition is fulfilled for a g-set E C G.

Proposition 2.24 Let G and M be Banach manifolds and 7- : E — ■ M be a

Banach bundle on M with fiber 1E. Additionally, let (1)

GxM E be a smooth G-parameter section, i.e. 1 is smooth and (Dg = 4) (g, .) : M E is a section in E for each g E G. Let 4) satisfy the condition: There is a countable trivialization {(U,0)}, 1,11 : Elu U X Eu, such that for each (U, t') • Ois a regular value of pr2 0 11) (I) :G x U • pr2 o 7,b o (Dg U

Eu and

Eu is a Fredholm, map with index r for all g E G.

Then there is a g-set E c G, such that the set Z9 = 4)g-1 (0) =1m E M I 4)9 (m) = 0 1 is a closed subrnanifold of M for all g G E. Another way of putting this would be to say: Given a generic parameter set E C G, the section 4)9 parametrized by g E E intersects the zero section in E transversally. We call the main result of this section the transversality result. The reader may notice that transversality in this sense and concerning the trajectory spaces is equivalent to the Morse-Smale condition, which is also defined in a way that describes a kind of transversality. The essential step of the proof is based on the following technical lemma, which can be proved easily by straightforward methods on Banach spaces.

Lemma 2.25 Let 4) be a bounded, linear map of Banach spaces, which is onto and of the form

4>:ExF—,G, (ke, f) = (Di (e) + 11'2(f ) Let 43.1 and 4, 2 be continuously linear and let (1)2 : F operator. Then there is a decomposition

E x F = ker

@top

such that H is a closed subspace of E X F.

H,

G be a Fredholm

2.3. TRANS VERSALITY

43

It should be emphasized that the proof relies crucially upon the Fredholm property of (I)2. If we could assume E and F to be Hilbert spaces, this lemma would appear to be trivial. Proof of proposition 2.24: Owing to the given countable trivialization of E we are able to assume without loss of generality that E is a trivial bundle. This is to say that we have a smooth map

(I):GxM-)E with respect to a Banach space E, such that 0 E E is a regular value and the maps tpg : M -> E are Fredholm maps with index r for all g E G. This assumption can be made because E = Eu is again a g-set if we consider the U countable trivialization {U,Ou}. Thus,

n

Z = cI) -1 (0) cGxM

(2.22)

is a smooth Banach manifold with associated tangent spaces given by

(2.23)

T,Z = ker D.:D(z)

for all z = (g,rn) E Z ,

as follows via local coordinate charts from Lemma 2.25 (!) and from the implicit function theorem. Now the main step of the proof is to conclude that the restriction of the projection map to this Banach manifold Z, 7r: Z -■ G , is a Fredholm map endowed with the same index as IDg i if we consider z -(g,m) E Z. First, let us focus on the kernels of the maps D7r(x) : T,Z - ■ E and D2(z) : Tm M - ■ E. We obtain the identity

(2.24)

ker D7r(z) -= T,Z

n T,M = ker D2(1)(z) ,

of finite-dimensional vector spaces with the assumption that D2 (z) is Fredholm. Second, we consider the following homomorphism between the at first merely algebraic quotient vector spaces:

(2.25)

Di (I) :

Tg G/R(Dr) [v]R(Dir)

—> 1->

EiR(D2(1)) [DOD v]R(D 2 4.) •

Having in mind the identities

v = D7r - (v, w)

and

/314) • v = -DA> - w

for each pair (y, w) E T,Z C T9 G x T,M, we immediately deduce the isomorphism property of i5ol. from the fact that D41(z) is onto. Knowing that the

CHAPTER 2. THE TRAJECTORY SPACES

44

has a finite dimension, D 1 (1. appears to be an isomorphism between finite-dimensional cokernels, cokernel of D2(z)

(2.26)

151i: coker (Dr(z))

coker D2(D(z) .

Hence, the equations (2.24) and (2.26) yield the Fredholm property of the projection map 7r. Finally, we are able to apply the well-known theorem of Sard-Smale to this smooth Fredholm map between Banach manifolds. It leads to a G-set E C G 6f regular values of 7r. Now let b E E. If (1)(b,m) does not vanish for any m from M, 0 is trivially a regular value of (Db. Therefore, let us consider an in E M satisfying (b, m) = 0. We then have to show that the operator (2.27)

D4)b(m) = D2cl(b,m) : Tr,M

E

is onto. Let -y E E be given arbitrarily. Due to the regularity of b with respect to 4, , there is a pair (a, 0) E TbG X T,M satisfying (2.28)

-y = atqb, m) • (a, 0) = D

a ± D2 43

• •

On the other hand, b is also regular with respect to r: Z G, any a E TbG we find an (a', 0') E T(b,,,)Z, such that it holds (2.29)

that is, given

a' = D7r(b, m) • (a' , 0') = a .

Hence, (a, [3') belongs to the kernel of D4)(b,m), i.e. (2.30)

0 = 1,4)(b,m) •

(a,

=

Di • a + DA • 0' -

Subtracting equation (2.30) from (2.28), we find a preimage of -y under the map D24(b, m) of the form y = D 2 4)( b, m) •(/3—

0')

•

yields the proof. It is the aim of this section to obtain a generic set of smooth Riemannian metrics, each of which satisfies the Morse-Smale condition. Within the given framework it is sufficient to fix an arbitrary metric go and to find a generic set of variations with respect to go. By this we mean a set of smooth sections A E End (TM) which are fibrewise self-adjoint with respect to go, positively definite and which differ from the section of identities by a sufficiently small amount. We then vary V f in terms of A V f. In order to be able to apply Proposition 2.24 to our transversality problem, we need a Banach manifold structure upon this parameter set of smooth sections in End (TM). This

45

2.3. TRANS VERSALITY

Definition 2.26 Let

be a smooth vector bundle on M endowed with a bundle norm I • I and a covariant derivation. Given a sequence (En).EN of positive real numbers, we define

IISlif = 1'0

Ek SUPM 117k Si

for all s E C cc (

(p) = max {IV z ,

This norm

)

=••

= 1, X E Tp M

IL gives rise to the vector space C'(0 = Is E C c'3 (01 11 8 ii, < oo .

It is easy to verify that (0( ) , IHI) is a Banach space if Cf() is nonvoid. An explicit construction of a sufficiently large set of smooth sections with suitably chosen compact support, which the reader can find in [FA gives rise to the following lemma. Lemma 2.27 There is a sequence (fn)EN such that the Banach space C€ (0 is dense in L 2 ().

Remark Obviously, the following inequality holds: if 0 < en en for all n E N .

(

In other words, we have a continuous embedding of Cf' in C. Summing up, we get the sequence of embeddings cont.

(2.31)

cont. ck cont.

C° for all k c N .

Definition 2.28 Setting Ego = Endsy,,,,g0 (TM) =

(p, T) E Ego I Tp positively definite w.r.t.

(*, *)p =

go(p)}

we now define the vector bundle Ego on M. It consists of the endomorphisms which are self-adjoint with respect to go , so that it forms a subbundle of End(TM) M. Additionally, 7 0 represents an open neighbourhood of the submanifold llm in Ego C End(TM). Due to the above remark C'W CN) is a continuous embedding and therefore we obtain

g90 =

I

(Ego) = {X E Ct (Ego ) Xp E (Tgo )p f.a. pE

OA an open neighbourhood of fi G Ce(Ego).

46

CHAPTER 2.

THE TRAJECTORY SPACES

Thus, g§,,, is a suitable Banach manifold of parameters with regard to our regularity analysis. It should be mentioned that each tangent space Tx ggo at X E gg. has the form Txg ga = C(E90 ) . In order to prove the central regularity result already for the time-dependent case which includes the time-independent one as a special case6 we still need a further restriction on the homotopy har3 : Definition 2.29 A finite homotopy ha 0 : R x M r, t —R fO, t R

h(t,.) =

R, i.e. satisfying

for some R

0,

is called regular, if it fulfills the condition: Given any R-critical point xo of h, i.e. Vh t (x0) = 0 for all t E R , the consequently well-defined operator from EX 0.TM

at + H2 ht (x0) : H 1 '2 (xTM) L

2 (4TM)

is onto. In this framework, H2 h1(X0) E End(Tx° M) denotes the well-defined (!) Hessian of ht at x0. Generally, this regularity condition is not fulfilled But if fa and P3 are functions with isolated critical points, we can always find a regular, finite homotopy between them. This is deduced by the following argument. Let us start from a non-regular R-critical point xo of ha° by assumption. We then choose a function k E CE,°(R x M, R) with suitably small support and satisfying dk t (x0) 4 0 for some t E R. Let us now replace the homotopy VO by h'`))3 = h'13 + k. This replacement can be iterated for all at most countably many non-regular R-critical points within a compact interval of time. We thus obtain a regular, finite homotopy. In particular, 11,0 is a regular homotopy if it is trivial, that is to say if ht = f ° for all t E R. Let us now study the following map: x

px1 : 2y

L2(px1 :12)

A V go ht •

(A,1')

;y+

V1 + filet = 0 for all t E R

Ihti 2

V90ht1 2

2.3. TRANS VERSALITY

47

Referring to the local coordinates and the trivialization of L2 (Pg) we compute the representation of 4> as (Bloc

ggc, x H 1-2 (-y*0)

L 2 (7*TM)

•

(A,)

+ 43() -;Y + \7 2 exP(e) 1 ( A Vht o exp,y

+

C°(7-90 ) is a continuous inclusion, 4>k), is also smooth with Since C€(Tgo ) respect to the parameter variable A. We may now state the decisive proposition, where we denote 4)1,3, again by 4' .

ggo ,

Proposition 2.30 Given any parameter A E (13 A

= (1) 10c(A,•) H 1 ' 2

(.y * (9)

L 2 (.-y*TM)

is a Fredholm map with the constant index i.i(x) - ii(y). regular homotopy, 0 E L2 is a regular value of (D.

Moreover, if tit is a

Proof. a) The Fredholm result is an immediate conclusion from Lemma 2.19 and the investigations in Section 2.2.3, because 43A E Frecl(y*TM, ‘y*V)

for all A E Ç90 .

b) The focal regularity result is now obtained as follows: The differential of 4' has the form D4 '(a,e) • (A 11) = Di(I)(A, ) • B + D24)(A, ) •77 -

Here, D2 4'(A,e) E E.-y •TM is a Fredholm operator of the already well-known type. For sake of computational plainness we choose the short notations d = D1 (13(A, ) • B E L 2 (7*TM) ,

d(t) = V2 exp(e) -1 (t) •

(BVht + Iht 1 2 (BVh t lAVh1 1 2 - AVht (AVht ,BVht))

+

l Ay ht 1 2 ) 3

together with a(t) = x(t) = (Vh t o

0 exp.),

(t)

e) (t)

y(t) = ((AVh t ) o expy e) (t) B(t) = (B o exp.), e) (t) .

48

CHAPTER 2. THE TRAJECTORY SPACES

Now let (A, e) E ggo x H 1,2 satisfy (13(A,e) = 0. Then e is smooth, too. Since due to a) R(D2(A, e)) is a closed subspace of L2 with finite codimension, the same is true for R(D). Thus, let us start from any vector c G L2 within the finite-dimensional subspace of L2 satisfying

(2.32) for all

( D(I) • (B,n), (:),„ _ 0 (B, ) E Ce (Ego ) X 11 1-2 (-y*TM) .

It remains to show that c(t) vanishes for all t E R. The relation (2.32) leads ' in particular to

(2.33)

( D2(I) -7/ , c), 2 = 0 for all „I E Ho .

As we already analysed in the Fredholm section, due to the smoothness e E C°°, this implies that c can be expressed as the solution of some non-autonomous linear differential equation

(2.34)

e(t) = X(t) • c(t)

with respect to a suitable trivialization of rysTM. If c is not the zero section, the uniqueness result for ordinary differential equations implies that c vanishes nowhere, i.e. c(t) 0, for all t E R. Briefly, given c 0, we obtain

(2.35)

(D1(13(A,e) - B , c) L2 = 0 for all B E Ce (Eg o ), c E C 00 (*TM) with c(t)

0 for all t E R .

Due to the above discussion about C6 this is generally true for all B E L 2 (E90 ), that is especially for all B with compact support. Choosing B with a suitably small support we may also give a pointwise formulation of (2.35) for a single t ER: Let us fix any arbitrary t E R. Then there exists a

a E TM,

0 with p = exp..r e(t) ,

satisfying the equation

(2.36)

( Bx + a (Bx (y, y) — y(y, Bx)),

a) = 0

for all B E End(TpM), which are symmetric with respect to (-,-) = go (• , .) on TpM. Let P be the symmetric operator Pz = z (y, y) — y ( y, z) . Then (2.36) may be written as

(2.37)

( (I1 + otP)Bs, e) = 0 for all B E Endsym (Tp M) .

2.3. TRANS VERSALITY

49

Since a is positive and P has a non-negative spectrum, it holds that

+ aP) Therefore, there is a z

(2.38)

0.

0 satisfying

(Bx, z) = 0

for all B

E Endsym (TpM) .

A simple computation shows that the space of symmetric plies with the property: For any pair y, Thus

endomorphisms com-

w 0 there exists a symmetric B with (By, w) 0 .

(2.38) gives rise to the conclusion

(2.39)

x(t) = (Vh f 0 expl, )(0 = 0

for all t E

R.

However, due to the identity (1,(A, e) = 0, it follows that s = expl, a constant trajectory. This leads to a contradiction, as we assumed regular homotopy thus implying the surjectivity of DA.

2.3.2

must be

ht to be a

The Regularity Results

Combining Propositions

2.24 and 2.30 we now are led to the result about

transversality. Theorem 1 Given a generic set of smooth metrics, the trajectory spaces M1, y and .A./t xh ,y are closed submanifolds of P1,2 x ,y of the finite dimension p(x) — Remarks. • A priori, we can only speak of local dimensions, because they are induced by the Fredholm indices of the local linearizations of the Fredholm section. These indices are only local analytic invariants which depend continuously on the underlying point of the zero set. The fact that they are equal on each component of this trajectory space follows from the mere dependence of the index formula on the fixed endpoints x and y in this special framework. In fact, if we look at the analogous Fredholm problem in Floer homology, we observe an additional parameter in the index formula, which involves a certain Chern number as a strictly local invariant, in general. In contrast to our situation, the trajectory spaces in Floer homology may have different dimensions for different components. • There is a corresponding result for the time-dependent case of homotopy trajectories, when the Riemannian metric is homotoped over a finite interval of time along with ILO. Let arbitrary generic metrics associated to the functions fa and f 13 be given. Then there is a generic set of associated homotopies gtaS, such that Mh.7 ,9`"8 has the property of a manifold.

CHAPTER 2.

50

THE TRAJECTORY SPACES aft

Finally, we have to study the case of A-parametrized trajectories: Let ho and VI/3 be finite homotopies of the form

to o (t,.)

f fa,

t

—R R

ff3 , t

'

. z = 0,1 ,

and let H° be a A-homotopy satisfying Haf3 : [0, 1] xRxM R Haf3 (A, t, ,)

f ;'

H°(0, • , .) =-

t R —R

.1r)3 (1, • , .) =

.

Then we consider the 1-parameter family of sections,

[0, 1]

x

pl,or2 y0

L 2(pzi,o2,y*07-m)

(A, 1,)1-4 ((t)+

(VHO(A,t,-)

,

o -y)(t) tER

with z a E Crit( fa) and y0 E Crit(f°). Corresponding to the first remark above, the Riemannian metric determining V may likewise be (A, t)-dependent. The following transversality result holds: Theorem 2 Let le and hr be regular, finite hornotopies together with its associated Morse-Snide metrics. Then there is a generic set of A-hornotopies HO of the above form and a generic set of suitable homotopies of the Riemannian metric, such that (Gaf3 ) -1 (0) is a (12(x„,) — (yo) + 1)-dimensional submanifold of [0,1] x 'Pzif,y0 Proof. Apart from the Fredholm property which is needed according to Proposition 2.24 with respect to the fixed local trivialization of the Banach bundle L2 (D1,2 * TM), the proof of this regularity follows in a way that is analogous ' x*,13 1/ to Proposition 2.30. And this Fredholm property is concluded as follows: Given the local trivialization at we consider the linearized map G

R x H1 '2 (7*TM) L 2 (-y*TM)

with

(7, ) 1-4 (G (A) • G2(A) • e) G2 (A) • = V + A(A) • e G1 : [0,1] —) L 2 (7* TM), G2: [0, 1]

Ery•TM

both smooth .

First, let us observe that the regularity assumption for the homotopie,s hoa s and tir implies that the operators G2(0) and G2(1) are onto. Second, G2(A)

2.4. COMPACTNESS

51

being a Fredholm operator for each A yields the finiteness of the codimension of R(G). Additionally, it is immediate from the continuity argument that the index of G(A) is constant on the entire interval [0, 1[. Hence, we compute the index of G by means of the index formula

ind G2 (0) = dim ker G2 (0)

= ti-L(Za) — i(y)

and due to the fact that G2 (0) is onto, as

ind G(A) = ind G(0) = 1 + ii(x a ) — ,u(yo) .

It should be mentioned at this stage that, generally, M.11 7:0 is a bounded manifold and that it is due to the regularity assumption at a À = 0, 1, that the boundary satisfies the relation

P 1,2 a./wH" Sa ,Y13 = m xH" o,,Y(3 n fo ,il x xc,,Y(3 Thus, there are no interior points u), E M which belong to the parameters = 0 or A = 1. For a schematic picture of this situation the reader is referred to Figure 4.4 on page 146.

2.4

Compactness

In this section we wish to study the compactness properties of the trajectory spaces in question. The fundamental feature which gives rise to the special results in this context is formed by the discrepancy between the 1/ 1 . 2-topology prescribed by the analytic methods and the CiTc-topology on the set of the solutions of the ordinary differential equation

=

—

Vf o-y .

Actually, this CiTc-topology appears to convey geometrical significance rather than the H"-topology. Due to the deduction from the implicit function theorem the finite-dimensional manifolds .A41 y and M.h o are obviously manifolds without boundary. Thus, in relation to the dimension one is naturally led to the question of compactness. In the case of relative Morse index ti(x) — p(y) = 1 this would immediately imply the finiteness of f71.1 y, and the case of dimensions 1 and 2 would allow an easy classification of the diffeomorphism type. As it will turn out throughout this investigation, the essential obstruction to the convergence of subsequences in the 1/ 1,2-topology is formed by the geometrical splitting-up into the so-called broken trajectories. Actually, it

52

CHAPTER 2.

THE TRAJECTORY SPACES

octopology appears to be apis in relation with the R-action on Mf that the Cr propriate for the description of this phenomenon. Due to the converse analysis concerning the gluing operation in the next section, the broken trajectories play the role of boundary points with regard to the trajectory manifolds. Briefly, we shall conclude the central results aimed at the construction of the canonical a-operator and of homomorphisms within the Morse homology from a kind of oriented cobordism equivalence for isolated, simply broken trajectories. 2.4.1

The Space of Unparametrized Trajectories

The aim of this subsection is to analyse the essential property of the trajectory manifold .)K y in the time-independent case, namely the symmetry with respect to additive reparametrization of the trajectories. This action is called 'time-shifting'. We shall henceforth use the designation 'time-independent' or `unparametrized' trajectories for the orbits of this time-shifting operation. Let 'y R M be a solution of the differential equation — f 07. Then, obviously, the same property holds for the shifted curve -y *7- =

7, =

r), as we compute

a

— (-y • 7- ) = at

+ T) •

7(11) = ( -y • r)(E) for all -y e .A41,10 that is the unparametrized set of points ry(N) remains unchanged. It is in this sense of Moreover, it holds that

point sets that we shall use the expression 'geometrical' behaviour. We make this observation of symmetry precise by the following proposition. Proposition 2.31 The additive group the manifold .A41 ,v by

Rx

R acts smoothly, freely and properly on —>

r)

-

•T=

r)

provided that the endpoints are distinct, x y, so that .A4f constant trajectories.

consists of non-

Proof. Notwithstanding the purpose of developing the Morse homology in strict analogy to Floer homology, that is to say by means of the Fredholm analysis which solely takes into account relative Morse theory, we shall build a bridge to classical Morse theory at this stage. Let us deviate here from the purely relative concept in order to elucidate the geometrical relations with the dynamical system which is provided by the (negative) gradient flow. This proof

2.4. COMPACTNESS

53

shall make use of the identi fication .A41,y Wu (x) smooth evaluation map

M

M)

E0 :

n ws(y). We consider the

'Y The smoothness of this map follows immediately from the representation by suitable local coordinates,

U(h(0)) C Th(0)M

E0,10 : H 2 (h*D)

(exp(DE0 o exph ) (C) = C(0) , where Eo,ice is continuously linear. If we restrict the evaluation map Eo to the trajectory manifold A/11, 9 , we obtain an embedding

E0 : A41,y

(2.40)

M

because the differential

DE0 (-y) : Ty.A/11, y —> T1 (0)M DE0 (-y) C = C(0) proves injective at each -y G M,f,ny . Specifically, let us consider a tangent vector C E T7.A41 = ker DF(-y) satisfying C(0) = 0. This argument analogous to (2.34) in Proposition 2.30, that C may be treated as solution of a linear, ordinary differential equation. Thus the identity C 0 is concluded from the uniqueness with respect to the initial value C(0) = O. As a consequence the evaluation map leads to the diffeomorphism between the manifolds

(2.41)

E0 :

jt4/4.,

w u (x) n WS (y) -y(0)

as it was already mentioned in the introduction. It is obvious that E0 identifies the group action (7- , '7) -y • T with the negative gradient flow,

x (Wu(x) n W 9 (y)) (t , p) This gradient flow restricted to Wu (x) proper ilk-action. Thus the equivalence

W u (x) n Ws (y) 41 4) .

n W 8 (y) represents a smooth, free and

-y • T = (E0-1 0 111 , 0 Eo) (7) yields the proof.

54

THE TRAJECTORY SPACES

CHAPTER 2.

In order to describe the orbit space .-A71-1, y = M 1. , y /R of the socalled unparametrized 7 or time-independent trajectories, we define the smooth function y; = f 0 Eo : A4-1, y —0 III with (2.42)

dy;(1) • e = df (1, (0)) • e(0) = (V f (ry (0)) , e(0)) .

Definition 2.32 Let f(y) < a < f(x), such that 'y(0) is not critical for any a regular value of (p and

'y E (p -1 (a). Then a.is

A4 .1,, dy

is a (p,(x)

- 1.1(y) -

= (p 1 (a) -

1) - dimensional submanifold of M 1 ,y and thus of 'Pl:, too.

From a geometrical point of view the orbit space .--A71-1 y represents exactly this transversal intersection of a surface associated to a regular level of f with the manifold Wu (x)nWs (y), which gives an equivalent description of the trajectory space A4-1 y as we have already seen. The reader is also referred to [Bo] and [S]. The following proposition yields an appropriate description for the orbit space. Proposition 2.33 The map Ta : R x

M-11/ -0 A4-1, y (7,1') 1-

-y • 7

represents a R-equivariant diffeomorphism, with respect to the trivial R-action on the left and to the action studied in Proposition 2.31 on the right side. Proof. Due to Proposition 2.31 the R-equivariance

= Ali a (7, 1') • CY is obvious. It is similarly straightforward to verify the bijectivity of V' by means of Proposition 2.31 and the uniqueness for solutions of ordinary differential equations. Thus it is sufficient to prove that

DANT,

ry) : R x T.y .A4 -1:!; --) 77-,,„M ,y

is an isomorphism. Let (s,) (2.43)

E R x T.y .A4 1:ay . -

Then (2.42) yields the identity

df (ry (0)) • (0) = 0 .

7This parametrization by the time variable t must not be confused with the dependence on an additional parameter A as will be considered below.

2.4. COMPACTNESS Given

55

(s,e) E ker Dtlia(0, -y), the

equation

0 = DkPa(0,-y) •

(s,

)

= • s+e

follows. From ;), = —V f o -y we obtain e(0) = s V f ('y( 0)), such that (2.43) implies the identity e(0) = O. Hence we conclude the injectivity of /A/a (0, -y) together with the surjectivity due to the dimension argument. Finally, we obtain the proof of the assertion from the identity tlia(7,-Y)

=

T

(Ta ( 0 ,1)) ,

because Or is a diffeomorphism according to Proposition 2.31. Summing up this geometrical investigation we obtain xi/a as a diffeomorphism

M-LY/liil M-1

;

which appears to be quite suitable in order to describe the differentiable structure on the space of unparametrized trajectories provided by the negative gradient flow of f.

2.4.2

The Compactness Result for Unparametrized Trajectories

First of all, let us state once again the fundamental assumptions for the investigation in the following text. Let M be a complete finite-dimensional Riemannian manifold and let f be a smooth function on M satisfying suitable conditions as studied above such that

= M-1,01 represents a (p(x) — p(y) — 1)-dimensional manifold. The first observation with respect to this statement is that the function f does not necessarily need to be a Morse function. It is sufficient for f to give rise to a transversal intersection of df and M as a zero section within the cotangent bundle T*M, and we need a metric which is generic with respect to the Morse-Smale condition. If we now wish to study the compactness properties of the trajectory manifold associated to f, we have to introduce further conditions on this function. In fact, we do not really need f to be a Morse function in the sense of coercivity, but to satisfy a more general condition as is fixed in the following definition.

Definition 2.34 We say that the function f fulfills the Palais-Sm,ale condition, which generalizes coercivity, if the following holds:

CHAPTER 2.

56

(P-S)

THE TRAJECTORY SPACES

Every sequence (xn)nEN C M, such that ( If(xn)1 )„ N is bounded and IV f (x n )I 0 converges, has a convergent subsequence.

A subset K c y is called compact up to broken trajectories of order y or up to (y — *fold broken trajectories exactly if for all (11 7,)„EN c K: Either *One,/ possesses a convergent subsequence, or there are critical points = yo,

, yi = y E Crit f,

and connecting trajectories together with associated reparametrixation times v i G Mf16,113+1'

= 0,

(rn,j)nEN C R

—

1,

such that we obtain the convergence Un k - Tn k ,j

for some subsequence (nk)kEN

Vj

•

GT..%

This convergence of trajectories wi w in CZy (rt, M) shall henceforth be called weak convergence, and the weak convergence of unparametrixed trajectories with respect to suitably provided repararnetrixation times is denoted by geometrical convergence. The geometrical convergence is sketched in Figure 2.1. Referring to these terms Yo

• Y2

Figure 2.1: Weak or geometrical convergence toward a simply broken trajectory the central statement of this section is given by

Proposition 2.35 If f satisfies the Palais-Sinale condition, the manifold M.1, y is compact up to broken trajectories of order p(x) — p(y).

2.4. COMPACTNESS

57

We are immediately led to the conclusion:

Corollary 2.36 If, moreover, it holds that p(x) = p(y) +1, then finite set of unpararnetrized trajectories.

is a

In order to give a proof for the central compactness result we first state an immediate consequence of the Palais-Smale condition.

2.37 If f satisfies (P-S), there is a constant depending merely on f, x and y, such that the set

Auxiliary Proposition Kr" = 1z E M

f(z) .f(x),

f(Y)

el

E>0

is relatively compact.

The following lemma shall express the 'geometrical compactness' of the unparametrized trajectory spaces in terms of compactness of the parametrized spaces with respect to the Cl' oc-topology. Lemma

2.38

Every sequence (u n )nE N C M1 ,u possesses a weakly convergent

subsequence

unk

vE

(R, M) .

Proof Essentially, the strategy for proving this lemma is to execute a transition from the H"-topology of the Hilbert submanifoldM.1 to the CL-topology. It appears to be of crucial importance that we have a uniform bound on the H1,2 -norm for the trajectories with fixed endpoints x, y E Grit f. The Cr,convergence will, in principle, follow iteratively by elliptic regularity.

The fact that the trajectories (u n ) E N arise from the negative gradient flow with respect to the fixed endpoints x and y yields the uniform estimate

(2.44) f

=

f

—

C7f 0 un ) dr 1(x) — f (y)

for all s t .

8

Let d be the Riemannian distance on M. Then the estimate d(un (t),un (s))

f

I itn (7- ) I

N/ It — Si

V

d7-

I

1141 (012 dT

\8 it

f (x) — f (Y)

CHAPTER 2. THE TRAJECTORY SPACES

58

gives rise to the equicontinuity of (un)nEN in C° (R, M). In order to be able to apply the theorem of Arzela-Ascoli we still have to analyse the pointwise convergence. At this stage the uniform L2-bound on the derivatives Un as well as the (P-S)-condition appear to be the decisive arguments. Thus let us choose a fixed to E R. According to Auxiliary Proposition 2.37 we can assume without loss of generality the inequality I (Vf 0 u n )(to)I Due to the asymptotic decrease Ern ItI (tr,)„EN C R, such that for all n E N: (2.45)

and

IVY (Un(t n))I = 6

f

for all n E N .

Vrf (un (t))1 = 0 we can find a sequence

I f (un (s)) I

for all s E [t, t0] •

f

Thus the auxiliary proposition helps us in fixing the points (un (tn )) nEN within the bounded set 1Cf ,Y. In other words, there is an ro > 0 such that the estimate d(un (tn ), x) < ro holds for all n E N. Referring to the inequality

d(x,un(to)) < ro + in (t0)

with

In (s)

=f

Iiin (r)1 dr

t„ the only step remaining in order to achieve pointwise convergence is to find an upper bound for (/„ (to)) nEN . Calculating

din —(r) = f 0 un( 7)i 6 ds d(fou) (7) = ° un(7)1 2 ds by means of (2.45) for tn T

to gives rise to the estimate

dl (7) (

) T9 c(i

(f 0 Un) (7)

Hence, we obtain the upper bound from

to i n (to) =

din --, (7)dr

d(f un) 1/ (7)dr =f (un(tn)) — f (un(to)) ds to

f (x) —

1(y)

Since (un (t)) nEN is a relatively compact set for each fixed instant t E R, the theorem of Arzela-Ascoli provides a subsequence (un k )kEN converging on

2.4.

COMPACTNESS

59

compact intervals. This amounts to stating that there is a E 03 (R, M) Un k I [—R,R]

y(t) = lim ?ink (t), t E R, s.t.

with

C° ([—R,RD

—>

k--poo

for all R

VI t—R,R1

O.

The fact that the trajectories (un k )kEr,i are solutions of the ordinary differential equation it = - V f (u) associated to the smooth gradient field implies the Ck convergence un k i[-R,ni

cku-R,RD

with ij =

v FR,Ri

y

iteratively for all k E N, thus concluding the proof.

0

Lemma 2.39 Let (un ) nEN C M .1 , y be a weakly convergent sequence vE

(R, M) ,

such that v E is a trajectory in the same trajectory space. Then the sequence (u) E N is also H 1,2 -convergent, Un

V

E M1,v .

Proof The first step of the proof is to observe that it is sufficient to show that the elements un of the sequence converge uniformly toward x and y, respectively, for Iti co. If this is true, Lemma 2.10 and the associated marginal on the dependence to = to(s) imply the existence of a fixed T 0, remark respect convergence with such that we observe uniform exponential asymptotic to local coordinates at x and y: (2.46)

cx ,t,e -A x I

lun,i0c

for I/

T and for all n E N .

Without loss of generality we can assume that the entire sequence lies within a coordinate neighbourhood of y in P, (Un)nEN C 01 '2 (v). Then referring to the local coordinates of the Hilbert manifold Un

= eXP, 6n/ n E Hit" (U s TM) ,

the weak convergence

o together with the uniform estimate (2.46) finally imply the asserted 1/ 1 ' 2-convergence. Thus we have to show the following uniform asymptotic convergence at y (and analogously at x):

CHAPTER 2.

60

THE TRAJECTORY SPACES

Given any neighbourhood U(y), there is an associated to E R such that the relation un (t) E U(y) holds for all t to and n E N. Let B(y) be an isolating, open neighbourhood of y, that is B(y)nCrit f = {y}. Then the (P-S) condition on f guarantees that

= fp E

MY)

0

if(P) — f(Y)i < f,

forms a fundamental .system of neighbourhoods of y. Moreover, we notice that U

= U un (R) nEN

is a bounded set. This is clear from the decomposition

fun(t)11V f 0 u n (t)1

U=

fun(t)

IIV f 0 u n (t)I <

nEN

nEN

In analogy to the proof of Lemma 2.38 the sets { un(t) uniformly bounded and

I

0 un(t)i f }

U { un (t) I IV f o u n (t)I <e C Kf' Y is relatively compact

are

.

nEN

In consequence of the boundedness of U, the vector field VI is globally Lipschitz continuous on U. This fact together with the equicontinuity of (Un)nEN from Lemma 2.38 gives rise to the existence of a constant c such that the estimate Vf 0 un (t) — IN7f un (s)i
1

holds uniformly for all n E N. Hence we conclude the inequality (2.47)

IV! ° un(s )1

>

f o u n (t)1 — Wit — si

for all s,t E R, n E N

Let us now prove the uniform convergence asserted above by contradiction: oc and If we assume that there is an c > 0 together with sequences nk due E N, then we first deduce, all k for satisfying Vf o u nk (tk)i > e tk oo

to (2.47), VI° unk ( 8

)1 >

E

for

2

and thus, second, tk+6 f(Unk

(4))

— f (unk(tk + 45)) = f

E.

E3

Ivf unk(s)1 ds > . — = 8c2 . 2

From this we extract the estimate 3 for all k f (unk (4)) — f (Y) > 8fc2

EN,

2.4. COMPACTNESS

61

in contradiction to the assumption of tk —* co and unk

V E .A4y .

Proof of proposition 2.35: At this stage we finally explore the obstruction to the strong H" 2 -convergence of the given sequence (u n ) nEN of unparametrized trajectories. As we remarked above, Lemma 2.38 leads us to the weak convergence v E Cc° (R, M)

(2.48)

and thus to a trajectory associated to the same negative gradient flow, o v -=- 0 ,

(— aat

but we cannot necessarily conclude that this eventually new trajectory lies in the same trajectory space .A/11, y . In fact, it is the R-invariance with respect to the parametrization by time which gives rise to the possibility of the so-called geometrical splitting-up of the sequence of trajectories. Given any convergent subsequence of parametrized representatives according to Lemma 2.38, the relation (2.48) implies the estimate

f (v(t)) c [f (y), f (x)]

for all t E R

Consequently, the set of possible manifolds of trajectories M, which might contain the element y, is already confined by the condition (2.49)

f(y) < f()

f(i) s f (x) .

Of course, we have to verify that y is an element of such a trajectory space y , anyway. We deduce this from (2.48) and the fundamental Palais-Smale condition as follows: First, relation (2.48) implies the confinement by f (y(T)) E [f (y), f (x)]

(2.50)

such that the solution property

= —V f o

for all T E R , y

gives rise to the estimate

(2.51) f 1V(s)1 2 ds = f (y(T)) — f (y (—T)) < f (x) — f (y)

for all T E

-T

Thus we obtain the finite L2-norm

f lid2ds<

ao and in consequence of this

the asymptotic behaviour (2.52)

lim Vf(v(t)) = lim V(t) = 0 . t—.+03 t—.±00

Finally, by combining the Palais-Smale condition on f with (2.50) and (2.52) we deduce the relation y E .A/q- , g with respect to critical points which satisfy the estimate (2.49).

CHAPTER 2.

62

THE TRAJECTORY SPACES

Now, we have to study separately the different cases for and Considering the first possibility, V E M y , we immediately obtain strong convergence from Lemma 2.39. All the Other cases which lead to the splittingup into broken trajectories, that is, without loss of generality, C

x,y

with 1(y)

f (y') ,

can now be controlled by means of suitable reparametrizations. Let us choose any regular level a of f satisfying 1(y) GaGf (y') and a reparametrization of (unk ), = Unk • rk = Unk (• TO, (rk)kEll C R such that the identity f ((un k rk)(0)) = a holds for all k E N. If we apply Lemma 2.38 once again to this sequence Uk, a further weakly convergent subsequence ft k —>

is extracted such that it complies with the estimate f(y)

f (î(+00))

f(

( -00))

(V)

•

Iterating these procedures of sorting by the values of f at the ends of the trajectories and executing suitable reparametrizations gives rise to a process which provides us step by step with either H 1,2 -convergent subsequences or merely CiTc-convergent subsequences tending toward constant, i.e. trivial, trajectories, or it 'recovers' new critical points of f. It is due to the Morse-Smale estimate for the Morse indices at the ends of the non-constant trajectories,

A (v (3) ( - oo)) - A(v (3) (+oo)) > 0 , that this iterative process of reparametrization stops at broken trajectories of order A(x) - A(y) at the latest. This can be seen immediately from the inequality (y) <

< / (v 0 )( - 00)) = ii(v 0 + 1) ( -1700)) <

< ii(x) ,

which necessarily holds after restriction to non-constant trajectories and suitable renumbering. Thus the assertion of the proposition follows. Remark It seems worth emphasizing, once again, that instead of the strong compactness condition for the sublevel-sets Ma(f) =f -1 (( which is meant by the coercivity for classical Morse functions, the Palais-Smale property defined above is thoroughly sufficient for the compactness result of the trajectory manifolds J+4, which plays an essential role in the definition of the a-operator.

2.4.

COMPACTNESS

2.4.3

63

The Compactness Result for Homotopy Trajectories

Essentially, the statements on the compactness of the trajectory manifolds for the a-operator may also be transferred to the trajectory spaces of the homotopy morphisms. However, since we restrict our considerations to Morse homology, we shall work out the following proofs for the compactness result under more specialized assumptions. This will simplify, for instance, the step of the uniform L2 -bound. Definition 2.40 Let us remember that the coercivity property of the Morse functions appeared as an essential part within the proof of the compactness result for unparametrized trajectories. Turning now to the time-dependent situation of a homotopy between Morse functions we have to guarantee an analogous kind between the of coercivity. We call a smooth homotopy h = h° : x A I —> two Morse functions f and r a Morse homotopy, if ht ( • ) = h(t, •) satisfies the following condition:

(2.53) lim ht„(xn) = 00, if ri—,

(tn, Xn)riEN C

R

X

M, d(x0,x n ) —> oo . 8

(X)

Now let us consider a smooth, regular Morse homotopy r f 0 . Apart from the demanded coercivity property it particularly holds that

ils o=

ah

=0

for all 1.51

R.

Thus, the (A(x) — (x0)) - dimensional manifold of the so-called homotopy trajectories (h-trajectory) comprises the solution curves uto of the ordinary non-autonomous differential equation (2.54)

;y(t) =

V'ht

o -y(t)

111 ± Iht12 1V ht1 2 with the ends fixed at the critical points x, and x o of fa and r, respectively. As to this manifold we are again provided with a compactness result similar to that in the time-independent case. Once again the only obstruction against the strong H 1 ' 2 -compactness consists of the CIZ-convergence toward broken trajectories. This time we additionally must distinguish between the R-invariant trajectories with respect to fc' and fO and the VO-trajectories. The convergence behaviour of homotopy trajectories is expressed by 8Note that this definition relies crucially on the completeness of the Riemannian manifold.

CHAPTER 2. THE TRAJECTORY SPACES

64

Proposition 2.41 Let (un ) EN

C M ilk" 0

be a sequence of hag -trajectories.

Provided that there is no 11 1,2 -convergent subsequence, there exist critical points =

, x („k E

Crit(f a ),

,X /13

=

Xs

E

Crit(P) ,

- ii(xp)

1sk+

and (hag -)trajectories . V c,E ivt • i+i, -0 -a mf's x. x 1+1,

yap E

1.3

x ,x a

together with reparametrixation sequences (Tai ,n)nEN) (TL)nEN C

R,

such that up to the choice of a respectively suitable subsequence weak convergence of the form Un • Ti

with 0 s j

a

Un• 0,n T

v 13

and tin cjn y ap

k - 1 and 0 j I - 1 holds. Moreover, the inequality 0

(X c,

<...<

(S,

0

11(X f3)

. . • < /.1# 0 1 )

is satisfied. Provided that we are able to prove a statement on the Crc-convergence in the case of homotopy trajectories, which is analogous to Lemma 2.38, the proof of this compactness result is accomplished according to Proposition 2.35. The weak convergence which proves crucial for this compactness result is obtained as follows:

Auxiliary Proposition 2.42 Each solution -y : (t- (to, xo),t + (to xo)) M of the non-autonomous differential equation (2.54) satisfies t+ (to , x 0 ) -=

that is the 2-parameter flow is global in forward time direction. Proof Let a maximal solution of (2.54) y: (t - , t+ ) -4 M be given. Due to the completeness of the metric space M and the requirement of coercivity (2.53) to the Morse homotopy haii it holds that lim h(t,-y(t)) = co , t-t+ if t+ is finite. Thus, provided that t+ is finite, it follows that (2.55)

d lim — h(s,-y(s))(t) = oc . t-t+ ds

2.4. COMPACTNESS

65

The computation

d

=

— h(s -y(s))(t) ds '

(t)) + Vht • ;y(t) ;

(254) .

(2.56)

IWit12

(nt

o -y(t)

+ lizt 121Vh t 12 together with (2.55) yields

.

lim ii(t,-y(t)) = t-t+

Applying the identity (2.54) once again we deduce

lim ;y(t)= 0 and hence a contradiction to the supposed maximality of

D

Auxiliary Proposition 2.43 There is a uniform 1, 2 -estimate for the homotopy trajectories with fixed endpoints. It is given as

I;y12 dt < c(x,„xp),

for all -y

E

M i/

n0

-00

Proof. First, we may estimate the integral by means of the identity (2.56). This yields

f I;d2 oo

f

co

dt

(2.54)

.

dt ....

-00 1+ Iht12 1V141 2 00

-00 (2.56)

--=

— f

f

00

I Vht1 2

IVht1 2

dt

_. 1/1+ ihti2 1Vhd 2

!li(r -y(r))(t) - it(t,-y(t))) dt

(-c

dr

'

-00

(2.57)

=

fa(x 0,) - f(x)

+f

h(t,-y(t))dt

-R

Since we deal with coercive Morse functions, the compactness of the set { s E (s } implies the relative compactness of the set { -y(-R) E M I fa (X) }. As a consequence the continuous 2-parameter flow from M I -Y E Auxiliary Proposition 2.42 yields the fact that

U -Y({- R, 10

is relatively compact .

CHAPTER 2.

66

THE TRAJECTORY SPACES

Therefore the uniform bound

= K(x„,x,3 ) (2.58)

=

sup { ih(t,-y(t))1

t

E [— R, R ] , Y E

Mxhno < oc

exists. Finally, the uniform L 2 -bound is obtained from (2.57) as

c(T,xp) =f (x) — P(x 0 ) + 2K R

Corollary 2.44 In particular, we obtain a compactness result for the h-trajectories as point sets from the auxiliary proposition together with the timeconstant Morse functions at the ends of the Morse homotopy: (2.59)

U 7(R) )E.M.ct

is relatively compact .

Relying on these preparations we are now able to prove the crucial weak convergence also for the homotopy trajectories. Lemma 2.45 Let (un ) nEN c M xh :i3,x0 be a sequence of 11' 13 -trajectories with fixed endpoints. Then there is a weakly convergent subsequence flk

V E C"(R,

.

Proof. The proof is accomplished by strict analogy to Lemma 2.38. Auxiliary Proposition 2.43 gives rise to the equicontinuity of the sequence. The subsequent Corollary 2.44 yields the pointwise convergence for approriate subsequences. It is worthwhile to mention that at this point we crucially relied on the coercivity of the asymptotically constant homotopy. The assertion now follows analogous to Lemma 2.38 together with the property (2.60)

yE

M7,o , with

P(4)

fa (x„),

(x p) f 3 (x) . 0

The proof of Proposition 2.41 is carried out now in accordance with principally the same scheme as for the time-independent trajectories. The only difference consists of the distinction between the invariant trajectories and the h-trajectories: Proof of proposition 2.41: It is sufficient to consider the case f(x) < f(x o ). Once again, let a be any fixed regular value from ( f (4), f (x ( )) and choose a reparametrization sequence in accordance with the condition (2.61)

2.4. COMPACTNESS

67

Obviously the divergence 7-1, -4 —co follows, that is, without loss of generality, Tk < —R. Since reparametrization leaves the uniform estimate from Auxiliary Proposition 2.43 invariant and since we can further aslume pointwise convergence of subsequences, we are able to deduce the weak convergence (2.62)

/in k

•

Tk

Vf

G

toward an r-trajectory up to the choice of a suitable subsequence. An analogous reparametrization algorithm thus leads us to the assertion. Corollary 2.46 Let us state an important conclusion from Auxiliary Proposition 2.42. Since the set {x E Mlf a (x) f"(x„,)} is compact, the same is true

for the set {7(+00 ) J f(—R))

r(x.), =

Vh °

}

due to the auxiliary proposition. Thus, each critical point x c, E Crit f" can be connected to only finitely many critical points x0 E Grit f via It'd5 -trajectories. In fact, this will be the reason that the hornotopy morphism 40" , which will be dealt with in Chapter 4, is indeed well-defined.

2.4.4

The Compactness Result for )-Parametrized Trajectories

The compactness result for the parameter trajectory manifold mr,f3y0 associated to a generic homotopy of Morse homotopies is obtained according to the same scheme as for homotopy trajectories. We recall that this manifold of Aparametrized trajectories (provided that we deal with a compact discrete set) forms the basis for the definition of the chain homotopy operator Va. Since the parameter dependence HA =

'), H : [0,1] x x M

is smooth and the parameter set compact, there are analogous uniform estimates. Once again, the only obstruction to the H' 2 -compactness occurs as the q" occonvergence toward broken trjaectories. Here, these broken trajectories consist of one H °-trajectory and several f"- and f'3 -trajectories depending on the dimension and the connected component of the trajectory manifold. In particular, we deduce the conclusion that M x11.10 represents a finite set ((A I , n ' ),. , (A,, u,,)} with A i 0,1, provided that we deal with the relative Morse index ti,(x a ) = p(y0 ) — 1.

CHAPTER 2. THE TRAJECTORY SPACES

68

2.5

Gluing

In the last section we saw how the broken trajectories associated to the timeindependent and time-dependent gradient flow form an obstruction to the compactness of the trajectory spaces. By introducing the gluing operation we now intend to complete the complementary concept of 'gluing' and 'compactness' as was described in the introduction. The gluing operation #p will map simply broken trajectories, that is, pairs of the form (u, ti) E M x

,y

X My,z

into the higher dimensional trajectory space M.,z in a way depending on an additional positive gluing parameter p. At first we restrict ourselves to the trajectory spaces of the time-independent gradient flow associated to a fixed Morse function. Principally, the construction of the gluing map for trajectory spaces in the time-dependent and .\-parametrized case is accomplished similarly with regard to the technical details. As to notation and the main ideas for the proof we follow the concept of gluing that was developed in [F6] for the symplectic case. For a compact presentation of this principle which in general is related to C. H. Taubes, the reader is referred to [F4].

2.5.1

Gluing for the Time-Independent Trajectory Spaces

The crucial problem concerning the gluing operation we are looking for is its dependence on noncanonical assumptions. For example, it appears that the compactness of the set of broken trajectories (u, y) E x M y,z , which are to be 'glued together' and mapped into M,,,, is an indispensable prerequisite. The crucial problem, however, is the lack of uniqueness. Notwithstanding the fact that given a compact set of broken trajectories we can always construct a gluing map, which embeds the glued trajectories in a unique way, this is performed in a necessarily noncanonical way. Clearly, any suitable composition with certain diffeomorphisms on the space of trajectories provides us with another, different gluing map endowed with the same convergence behaviour with regard to the compactness-gluing cobordism. A consequence of this uncertainty is the lack of a natural, immediate law of associativity,

(u#,

w = u#,„ (v# 2 w) .

The result, which will be sufficient according to our concerns with homology theory, is stated as follows:

2.5. GLUING

69

Theorem 3 Given a compact set of simply broken trajectories K c M1 ,y x .A4 , there is a lower bound pK 0 and a smooth map : K x [pK , oo) (u, y, p) 1—> u# pv

#p K M1, z is an embedding for each gluing papK . Moreover, given a compact set k C x of unparametrized trajectories, # induces a smooth embedding satisfying: The map

rameter p

î:kxr such that we obtain weak convergence toward the simply broken trajectory

(11,b) as p tends to infinity, p —> co. On the other hand we obtain the converse result that any sequence of unparametrized trajectories converging to a simply broken trajectory finally lies within the range of such a gluing map #. The gluing procedure may be sketched as in Figure 2.2.

Figure 2.2: Gluing for unparametrized trajectories

In relation to the investigation in the last section this theorem supplies us with a far more detailed characterization of the obstructions to compactness of the unparametrized trajectory spaces Mx at least as concerns the relative Morse index 2. Actually, these obstructions are described topologically as cylindrical ends k x [p oo) with the set k of simply broken

70

CHAPTER 2. THE TRAJECTORY SPACES

trajectories as a cross section. It should be mentioned that a treatment of higher dimensional trajectory spaces would require a more involved analysis. But in principle we are led to a kind of cobordism theory as we have already explained in the introduction. This is what we call the 'compactness-gluing'

cobordism. The construction of the gluing map is carried out in three steps: First, we establish a smooth pre-gluing map #p°, which glues broken trajectosuch that the ries (u, y) E M x , y X Al y,z rendering a smooth curve in image (uepy)(R) differs from u() U y(IA) by a sufficiently small amount and, moreover, such that the convergence toward the unparametrized trajectories u UiT pV

0.

‘

kU, V)

is provided for p oo. The crucial second step will provide a procedure of associating nearby trajectories from .A/G,, to these preparatorily glued curves, which represent approximate solutions of the negative gradient flow, in a unique way for large gluing parameters p. In order to guarantee uniqueness we in principle use a normal bundle of .A4 x ,z within P4'!. However, this normal bundle is not supplied canonically. By means of the projection along the noncanonically transversal fibres° we find a contracting map. Due to the Banach contraction mapping principle and in consequence the implicit function theorem, we obtain a unique solution of the negative gradient flow if the gluing parameter p is large enough, that is to say that we differ little enough from the original trajectories up to time-shifting. Third and last, we have to verify the embedding property. This is largely owing to the uniqueness'° guaranteed by the Banach fixed point theorem, if p is large enough.

The Pre-Gluing Map and the Uncanonical Normal Bundle Let us assume as in Theorem 3 that K C M1, y x i‘rt-, z is a compact subset. Then given any pair (u, y) E K, we choose smooth curves (ii, E C" x,IL X C" yz appropriately such that within the Banach manifolds Px1, and Py': 1. there are coordinate neighbourhoods Lk and lib containing the previously given trajectories: E /1,-, and V E tii,- . In particular, let us choose Ii and endpoint y:

asymptotically constant at the common

ii(t) = y, t T jj(t) _ y, t _7, , for some T = TO, 9In fact, this can be formulated as a retraction. lOstill referring to the noncanonical normal bundle

.

2.5. GLUING

71

Due to the compactness of K we find a finite covering by such neighbourhoods X 145 yielding the constant

14

f)) .

T(K) -=

In the following chapter, on orientation and gluing with respect to Fredholm operators, especially such asymptotically constant curves will play a significant role. These curves allow us to define the simplest kind of gluing operation," namely ( il#p i3)(t) =

(2.63)

(t

13) ' p),

t 0 t o

We thus obtain

ft# pf, E

CZ

for all p T + 1 with T =

i3) .

We shall henceforth use the symbol # for the gluing operation concerning trajectories as well as for the gluing of asymptotically constant curves.

Definition 2.47 Throughout the whole section we will use the following cut-off functions:

which are defined by

f

(t)

1

,

= 1 0,

t-1 t o

and (3+ (t) =

1,

t

0

t1

We are now able to define the preliminary, approximative gluing map, that is the so-called pre-gluing map,

#°

K x {po,) -4 Pl:v2 u#''pv (u, y, p)

by

(2.64)

u#:)v = exp,-", pf, (0 - [exp,V (u)] p 0+ • [expi l (v)] _p)

for po = Po(K) > T(K) +1. Here, (ft, f;) are chosen suitably with (u,v) E /A X Lif, referring to the above covering. The notation u p (t) u(t + p), t E describes the time-shifting that has been already studied above. 1 1Note that this does not concern non-trivial trajectories of the gradient flow.

CHAPTER 2. THE TRAJECTORY SPACES

72

It is easy to verify that the definition of urpv is independent of the individually chosen coordinate charts from the fixed atlas, as the equivalent representation

up (t),

t

—1 t), It I 1 t 1

expv [0- expi7 1 (up) + 0+ exP; 1 v_ p (t),

(2.65) (u# 0°v)(t)

follows from the asymptotic behaviour of it and 1 and the definition of Ofpf). Hence it is only with respect to the lower bound po(K) that the definition of #° depends on the choice of the covering of K. Moreover, the two equivalent representations (2.64) and (2.65) yield the smoothness of this pre-gluing map r. We conclude the differentiability with respect to u and y from (2.64), and formula (2.65) implies the differentiability with respect to p yielding the

formulal2

t

i(t),

(2.66)

4 (ur„v)(t) =

0 — V2 exP; 1 • 4P I V2 exP y • [_ 0+ V 2 exp17 1 * i/—p

(t) , Itl ..,., 1 t ?.. 1

—7)_p(t), E L2

—1

((u# 0° 0 *TM) .

Remark As we will see in Proposition 2.50 below, already this pre-gluing map #° has the embedding property if p is large enough, i.e. p 0(K) for 0(K) large enough. This can be seen from the fact that the differential associated to the map (2.64)

ker DF„ x ker DFv

()

H 1 '2 ((urpv)*T M) 1-4 V2 exp (0 — (V2 eXP

- p 0+

(V2 exP -1

will prove injective for large p. In the following considerations we provide the fundamental structures which will serve to find the actual gluing map according to the Banach contraction principle. We shall construct suitable Hilbert bundles, a linear gluing version, which is injective, and a normal bundle which is to guarantee the uniqueness of the gluing operation. It is worth mentioning that if we could cover K by only one single pair of coordinate neighbourhoods, we would be able to reduce the whole problem to the trivialized framework. Then we could renounce this somewhat extensive bundle constructions. The trouble in the non-trivial case, however, 12For the definition of

V2

exp the reader is referred to Definition A.15 in Appendix A.2.

2.5. GLUING

73

consists of determining the correction term between the actual gluing of trajectories and the already defined approximate pre-gluing in a way which is independent of the trivialization. Actually, this amounts to the below construction of a normal bundle L. Throughout the following investigations we will use the notations

x = (u,v, p) E K x [po, oo), wx = u #po v E pz,z = px1 :2z, : Px,z L 2 (P;,,TM) + Vf o , where we refer to the local representation with respect to w E Cx".: (2.67) F = V2 exp 1 oFo o expw : H 1 '2 (w*TM) c Ow L 2 (w*TM) . Since the broken trajectories (u, y) E K are composed of smooth curves, the nonlinear operator Foy), E Fred(wx*TM) is well-defined as well. The Riemannian metric ( ) on M induces well-defined Riemannian metrics Ic

)9./2

( e,

) ,?y,2

for

-y

WO,

=

•V't) Vt(

=

E

(

) ,(2/ 2

C C

together with the associated fibre norms (see the appendix) H1,2

),y (t) dt

11 11 1,2 and

ill! 0,2 on the bundles

(nz TM) c L 2 (P1,z TM) . K x [P0,00 ) —+ Px,z supplies us with the Hilbert

Thus the smooth map bundles

H = #°* H1 '2 (7);,z TM) and L = #°* L 2 (n z TM) on the base space K x [p0, x). Additionally, the inclusion j : K c-4 M x ,y x M v , z induces the Hilbert bundle (H1,2) 2

i*

X H1ljet v, )

on K x [pc) , oo). According to these bundles H and L we are able to identify the map F,„ from (2.67) for wx E (K x [po , oo)) via the pre-gluing map # 0 : x 1-4 wx with the fibre restriction of a smooth bundle map

CHAPTER 2. THE TRAJECTORY SPACES

74

HO

L

In this context 0 denotes an open neighbourhood of the zero section, such that the boundary is properly bounded away from the zero section with respect to the norm 11 . 111 2 - This is concluded by means of the injectivity neighbourhood associated to the exponential map, the compactness of K and the construction

of the pre-gluing map, because the image of the curves uepv ( R) remains within a compact neighbourhood. This bundle map F:HDO —t L together with the identification Fx ---- Fwx thus has the explicit representation {

V2 eXP0 up ( -1 - [( -g + V f) 0 eXpup fj (t),

t -., —1

V2 eXP_ p (C) -1 - {(A. + V f) 0 expv_ , e'i (t),

t „--. 1

{ Flip W)(t),

t -<, —1

Fv _ p (C)(t), t ? 1 with "

E E E H-11,2p

111,00)

t,'

and we obtain the identity

= Fo (wx ) . Moreover, the fibre derivative

D2F(x)(0) = DF(0) : Hx Lx at 0 is exactly the linearization of Fo , which has been already studied as a Fredholm operator of the type

0

+ X (t)

We shall henceforth use the shorter notation

Dx = DF(0), xEKx [po, co) Definition 2.48 We are able to define the linear version of the pre-gluing operation as a bundle homomorphisrn by

2.5. GLUING

(C#x()(t) =

75

V 2 exp [ +

0— V2 eXP -1

0-1- V2 exp -1

<- p

The fibre derivative V2 exp is taken at the points exp; 1 (uepv (t)) , exp y-1 (up (t)) and exp 17 1 (v_ p (t)), respectively This corresponds exactly to the linearization of e as it is given by equation (2.65). The crucial point of the construction of the gluing map can be found in the process which deduces the linearization of the trajectory gluing from this linear version of the approximate gluing #°. By the linearization of the actual trajectory gluing we mean a projection onto the tangent bundle of Mx,z as a subbundle of H 1 '2 But it is exactly this projection map which cannot be canonical within our construction. We need the concept of orthogonality, which is global with respect to K, namely the normal bundle of Mx , z as has been already described above. So, let us consider the following candidate for a normal bundle: Definition 2.49 Since D. = DF„ and D„ = DFL, associated to (u, v) E are surjective Fredholm operators,

Mx,y X My,z

(ker)2 = U ker Du x ker

C (H 1 '2 ) 2

xEK x [Po oc) is a subbundle with constant and finite dimension. Then the following vector space is well-defined for any xEKx [po, oc):

Lx± = vx E H 2 = Hx I {

(vx, C#p()' 2 = for all (C, () E ker Du x ker Dv }

The following first, fundamental proposition within this discussion of the gluing problem states that these vector spaces represent the fibres of the normal bundle we are looking for.

CHAPTER 2. THE TRAJECTORY SPACES

76

Proposition 2.50 There is a lower parameter bound pi po, such that for any gluing parameter p pi and broken trajectory (u, V) E K the Fredholm operator Dx : Hx L x is onto and the composition of #x with the L 2 orthogonal projection onto ker Dx induces an isomorphism )

Ox = Proj kLexi. Dx 0 # x : ker Du x ker Dv

ker Dx .

We deduce from this first result that ker D =

ker D x

xEKx[pi,c0)

is a subbundle of H on K x that

oc) with the finite dimension p,(x)— p(z) and #1(ker)2 : (ker) 2 —* H

represents a bundle homomorphism. Hence, we obtain Li

= U xEKx[pi,co)

as a finite-codimensional vector bundle on K x [pi , oc) satisfying the relation etop R (#1(ker) 2 ) = H • In fact, as we shall see below, we are led to the decomposition (2.68)

L I etup ker D -= H .

Summing up these arguments we may say that LI can be considered to be the normal bundle which is needed for the gluing construction.

Proof of proposition 2.50: Concerning the central feature, the proof of this nonlinear gluing theorem is essentially patterned on the proof for an analogous gluing operation in Floer homology as can be found in [F-HI. As we already noticed above, the operator Dx = DF„x referring to wx = uepv E is a Fredholm operator satisfying the identity ind Dx = p(x) — p(z) . According to the assumption that the Morse-Smale condition be fulfilled, the Fredholm operators D. and D, are onto. Hence we obtain the expressions dim ker D. = ind D0 = p(x) — dim ker Dv = ind Dv = p(y) — p(z) .

2.5. GLUING

77

Therefore it is sufficient to show that, given any broken trajectory (u, y) E K, we can find a lower parameter bound p(u,t,), such that Ox is onto for all p p(u, v). This is due to the assumed compactness of K which implies a uniform constant p(K), such that Ox is onto for all xEKx [p(K), oc). Hence we can conclude dim ker Dx 1.<., dim ker Du I dim ker D, - -

(cf.)

= p(x) - p(z) = ind Dx dim ker Dx ,

which gives rise immediately to the identity dim ker Dx = dim ker Du + dim ker Dv and which consequently implies the asserted isomorphism property of Ox for all xElCx [p(K), co). Therefore, let us choose an arbitrary, fixed broken trajectory (u, y) E K. Then it suffices to find a constant C <0 and a lower parameter bound such that the uniform estimate

(2.69)

IID x II L %CIIIILz

holds for any p

/3 and e E

L. This is clear, because there is a

e E kerDx with (e, R(# x ))q, = 0 , if 45x should fail to be surjective. But owing to (2.69) this e E L x± has to be the zero vector. Let us now prove the estimate (2.69) indirectly. We assume sequences (on)nerl and satisfying the conditions

(er,),EN

(2.70)

pn —> 00,

en E L, gn1ILL

=1

and

11Dx„n11 -40

First, we wish to deduce the convergence

11,1c:2

en —c*

from these assumptions. As a matter of simplification we shall transfer the situation to the trivialized case of li n . Because we have chosen the broken trajectory (u, y) E K to be fixed and due to the fact that, provided any large gluing parameter p, the piece of curve urou-1,1] lies within the normal neighbourhood N(y) of y with respect to the exponential map, we obtain

U (UlepV) * TM Pe [Po '03)

CHAPTER 2. THE TRAJECTORY SPACES

78

as a trivial vector bundle onliTt x [po , oo). Thus starting from a trivialization (Dy : TM1N( y ) —■ N(y)xRn we find further trivializations (D u and (I),, of u*TM and v*TM respectively together with

U (ulepv)*Tm

[P0,00) x x

PE [ P o ,co )

satisfying the relations 4'1[1,00)

= 1%111,00) and 41 (-00,-11 =--= (1) 1,1 ( - 00, - 11

such that we obtain the uniform estimate 13 sup I#v (t)l

0< ci and for all p

c2

for all p

Po ,

P

tER

po the isomorphisms

(kx Hx1,2 _f_) 4 H1,2

liC )

114>x11 0,

with 0 <

C2

•

Concerning the class L 2 we are led to an analogous result. Next, these uniform estimates enable us to consider assumptions (2.70) above without loss of generality for the trivial case

0 D Xn

A Xn H 1'2 (R) Rn )

L2 ( 11 1 Rn )

that is (2.71) (en)neN C H 1 ' 2 (R,Rn ), 1442 =1 with IlDxnenh2 —* 0 . Here, Axa is a curve AXn E C"(rk, End(Er))

determined by the trivialization 4:13 x„.

Referring to the trivialization

TM Rn we obtain Ay = 4>y • D(V2exp y--1 .(Vf o exp y )) • as a constant operator which is conjugated to the Hessian H 2 f(y), so that the Fredholm operator

induces an isomorphism (2.72) 13I•1•p l is the norm of 4)p : TM upon TM.

: H 1 ' 2 2--% L 2 . aln with respect to the Riemannian metric (•,•)

79

2.5. GLUING

This is immediate from the trivial spectral flow associated to the constant curve A. Now let /3: R [0, 1] be a smooth cut-off function of the type

r

It'

1, = 0,

p„,. Additionally, we choose the short

oc be the sequence rn =

and let rn notation

Or' (t) =

rn

, s.t.

1• t d — 0( — ) • rn )(t) = — (O dt . rn rn

These preparations give rise to the following series of estimates:

▪

liDy(O rn •

en) L 2

(0;'. ) •

+ Or '

rn 1

P(

• Dign

L2

_t35 ). 71 L2 + liDgnIl L 2()' c = [rn Dxn )en II

•

— rn • + iip xnenii L2 +

▪

sup HA(t) — rn • + IIDIlz +tE[—r n ,rn ]

1

°I

(Dy

ilEnclOr9

Rn 111,2

Due to assumption (2.71) and the estimate sup llAx,, — A

[—r, ,rn j

EndOV

— Ay ll

sup

tE[--1,11 +

sup IlAn (t + pn ) — Ay !!

tE1—r„,-11

±

SUP

(t —

pn ) — A y ll

,

tE[1,rn]

where all terms on the right side tend to zero as u and y converge exponentially to the critical point y, we conclude

WI/ (Or n • en)II L2 =0

•

Therefore, the isomorphism Dy from (2.72) yields the Hilo'c2 -convergence we wish to verify. Finally, this result can be transferred back to the non-trivial case, that is

(2.73)

en

0.

become more and more concenBy this conclusion, that the supports of trated at the asymptotic ends as n increases, we are now able to transfer the estimate

11Dxnen1I LL,

0

CHAPTER 2. THE TRAJECTORY SPACES

80

for the varying Fredholm operators to the fixed operators Du and Dv :

1011- Pn en' -P'l Pi-en 11 .Pn + 110T L2

111(V._ impuen,- p,,11 14, Dup jfIIILUpn

1 /317 D Xnn 11 13Xnen lILL

(r-2,-1 , 1)

-

According to assumption (2.70) and the above estimate (2.73) we are led to the convergence

0• Since D u is a Fredholm operator, this amounts to the existence of an a E ker Du with 1-* 2

0 1—pn en

P n ---4

a

.

In particular we obtain the convergence

11 0T • Gi In an analogous manner we are provided with a "f E ker Du satisfying the relation

11 )ti

— ^Y— Pnlig_ pn

•

Summing up, we observe the following approximate behaviour:

1 01 en —

(2.74)

E ker Du ,

CtP.IIL2

E ker Do .

110-±iGt

The contradiction with the former assumption is now caused by trying to piece together all these different conclusions in the series of identities:

1

gn11L2x7, lin (fiTeu Otien ± (1 n—oco

=(2.73)

lip

(01

en, en

)LL,

- Ot1)en7 en )qc „

(Ot1en7

en )LI n

as supp(1 - 017 -3) C [-2, 2] (2.74)

-

lip ( 0 -0/p,‘ 7 en )q

-

liM

-

0, as G, E Lx± for all n E N .

Ox.7,

(0+ '"Y—p, 7

G. )LL‘

en 4Z,,

The proof of the assertion follows.

D

81

2.5. GLUING

The Trajectory Gluing Up to now we have merely analysed the pre-gluing e as an approximate version of the gluing operation. However, the thus defined curves wx = u#pc'v are not yet trajectories for the negative gradient flow. Resorting to the normal bundle L i constructed above we now try to find the correction term for e with respect to the actual trajectory gluing. Moreover, this correction term has to be provided globally on the entire base space K x [pi , oo). This requires K to be compact. In other words, our aim is to find a 'correction section' -y , oo) in a unique way, so that in the bundle H on K x exp

'y()

for xEK x [pi oo)

finally represents a gradient flow trajectory. That is to say that we are searching for a section ry:Kx [pb oo) 11 satisfying Fo -y= 0 with respect to the formerly defined bundle map

HO

L

K x [po , oo)

Additionally, we want exp -y(x) to comply with the geometrical convergence toward the broken trajectory (u, y) as the gluing parameter p tends to infinity, p co. Since we are provided with the convergence property for uepv due to the construction, we shall be satisfied with a sufficiently rapid convergence for the correction term, -y (uep v)

as p co .

We shall derive all these results from the existence and uniqueness theorems which are supplied within the Banach space calculus. In this framework the bundle L I guarantees the (noncanonical!) uniqueness. It is essentially the fixed point calculus due to a contraction mapping which will lead us to the necessary correction term. We will obtain this contraction mapping from Proposition 2.50, that is the bundle decomposition (2.68), which gives rise to the vector bundle isomorphism DI :

'=% L2 .

CHAPTER 2. THE TRAJECTORY SPACES

82

Thus, considering the inverse map 14

G : L2 —.‘ ■ L i , G = (DI L ±)-1 , we can extract the following estimate from a refinement of the proof of Proposition 2.50: Lemma

2.51 There is a lower parameter bound p2 po and a uniform 0, such that the isomorphism G satisfies the estimate

constant CK,i >

liGx6111,2,x

C K ,11161lo,2, x for all X E K x [p2 ,00), 6 E L2x .

Proof The decisive feature within this proof is the uniformness with respect to the broken trajectories (u, y) E K. An equivalent statement of the assertion via ( = Gx - 6 E L x± is given by: ii(Iii,z,x

for all ( E Lx-L , XEKx [p2, co) .

CK, 1 IID • (110,2,x

In order to carry out an indirect proof similar to Proposition 2.50 we assume a parameter sequence pn oo and associated broken trajectories ((un, vn)) flE N C K together with E L, Xn = unepn vn satisfying the conditions (2.75)

11(7111 1 ,20( n= 1 and

•

In contrast to Proposition 2.50 the main difficulty lies in the estimate for an inconstant sequence ((u.„, vn ' )) nEN instead of a fixed pair (u, y). Therefore we construct an appropriate transformation to the situation of such a fixed pair. Due to the compactness of K, we can assume without loss of generality an H 1,2 -convergent sequence, that is in particular a C° -convergent sequence

(un , v)

Co

(u,v) E K .

Considering the contractability of the curves u, y and ury we find an open neighbourhood U of u(liTt) U v(rt), a trivialization (I) : TM! u an no E N, such that the inclusion

Xn,m(K) = u n #p%vn (R) C U holds for all m, n

U

X

Rn and

no .

In order to transfer assumption (2.75) to the fixed broken trajectory (u, y), we consider the following homomorphism family between vector bundles on Ft 14The reader is referred to [F6] or [F4] in the case of Floer homology.

2.5.

83

GLUING (un #Ly n )*TM

(uK,„y)*TM

relying on the definition

(1)rn,n

(Wpm v(t), Pr2 ° (13 (X,.,m(t))) •

(x„,m(t)) = 13-1

Let us henceforth use the notations Xnn = unep„vn and xn = u#y. Then the explicit construction of the pre-gluing map irp together with the H" 2 -convergnce (nn, Vo ) 111'2 (U, V)

yields the convergence of the induced Banach space isomorphisms n

with

(8)(t) =

. H 1,2 -2, H1,2 an d L2,, Xnn

L2

respectively,

srtinn (S(t)). This is due to the convergence

and

11 4)nilr(H 2 0H) 11(1), o D x.nn

as n goes to infinity, n

—

OnlIG(LL L ,LL) 0

d'n

1

)

Do.

Moreover, given any c E (0, 1) the convergence 1),,, an. no(c) E N, so that we can transform the former assumption

=1

provides

for all (in E Lx±r„,

into the estimate 1—c

(PrLL ° (N ) (")

1

1,x n,

+E

for all n no. Piecing everything together we come to conditions similar to those in the proof of Proposition 2.50. Now confined to the fixed pair of trajectories (u, y) and using the notation = (Pri,L o (1)n) ((n ) E

we obtain the conditions

Rnili,x „

IlD

II

■

—

Pn

,

CHAPTER 2. THE TRAJECTORY SPACES

84 as

(Pr"x - ) ° (1)n ) ((n)

Lan

Hence we are finally able to conclude as in Proposition n—■ co

2.50:

Ile.1 1 0,x „ = o I1Dx ,GLii o , xn -4 0 leads to the estimate

On the other hand, the convergence

C lim ilen il oocn

lini II

110,X n

for some suitable constant C> 0. But this would imply the inequality

(1 + C) l im lim lienlli,x„Rn110, xn. = 0 contradicting 1

-

c

lim Ilenlli,xn

as stated above. Thus we have proven the uniform estimate of the assertion. Let us now introduce the zero point lemma which is central within our space calculus.

Banach

Lemma 2.52 Let f : E -4 F be a smooth map between Banach spaces of the form

f (x) = f (0) + D f (0) - z + N (x)

with

dim ker Df(0) <œ

Let us further assume G : F E to be a right inverse,

D f (0) o G = idF and let C > 0 be a constant such that the contraction property

IIGN(x) - GN(y)II

C( 114 + 11Y11 ) 11x - Yll

is satisfied for all x, y E B (0) with e = Then the initial condition implies the unique existence of a zero point of the map f iiG f ( 0 )II .

E B,(0)

n G(F) ,

which additionally fulfills the estimate

lixoli

2 IIGI(0)11 -

85

2.5. GLUEY G

Proof. The proof is accomplished as a direct application of the contraction mapping principle. Considering the mapping

: E —> E , (p(x) = —G (f (0) + (x)) , we easily calculate the inclusion relation, as x E B,(0) yields the estimate

1150(x)

2

1+1)< E Ce2 = e ( 12 + Cc) = f ( 2 5

due to the assumption. Moreover, cp contracts on B,(0) as we immediately see from

=

II(x) — So(Y)li

IIGN(x) — GN(Y)11 2 C2€ lix — Yii = g Ilx — Yil —2

-

Hence we find a unique xo E B,(0) satisfying the identity (p(x0) = xo. Additionally we obtain the estimate

iixoli

ii(P( 0)11

licP(x0) — LP( 0)11

—21 IxoU•

Thus we deduce the inequality Poll <2 f (0) II and complete the proof. It is by means of this fundamental lemma that we shall later prove the existence of the necessary correction section -y : K x [pk, oo) —> L'. Let us choose a fixed XEK x [p2, co) and let us consider the smooth mapping between the Banach

spaces

Fx Hx1,2 L2x

that is, explicitly, 15

Fx (x) = Vtx + e(x) /bx + V2 eXP(X) -1 (Vf o exPwx with F(0) = F(wx ), DFx (0) = D. Considering the expansion

F(x) = F(0) + D(0) x + Nx (0, x) we compute the difference of the nonlinear terms as

Nx (x) — N x (y) = Fx (x) — Fx (y) — D x (0) • (x — y) = (Dx (y) — Dx (0)) (x — y) + Nx (y , x — y) . 15For the notations used in this formula the reader is referred to the discussion in Appendix A.2.

CHAPTER 2.

86

THE TRAJECTORY SPACES

Analysis of the nonlinear dependence in the smooth exponential terms together with the compactness of K, that is

(2.76)

U wx (ITZ) xucx[po,00)

relatively compact ,

yields an estimate which is uniform with respect to x,

(2.77) IIN x(T)

N (V)I10

C K +2 (II X II1,x

IlY111,x) IIX

YI11,X

The constant C K ,2 is a function of the C2-norm of V2 exp on the compact closure of the set in (2.76). The required estimate thus follows from this inequality together with Lemma 2.51 and with CK = CK,1 • CK.2,

IIG x N x (X) — GxNx(y)111,x (2.78)

and 11Gx 1'x (0)11 1,x =

Ylli,x

IIGxF(wx)111, x

C

11F(wx)110, x

for all x EKx [p2, oo) . Finally, it remains to show that the deviation of the approximately glued trajectories from the actual gradient flow trajectories tends to zero as the gluing parameter p goes to infinity. This follows immediately from the next lemma.

Lemma 2.53 There are uniform constants m> 0 and c> 0, such that the estimate a e- mP ilF(wx)110,x holds for all

p

Po and (u, v) E K.

Proof. Due to the construction of the pre-gluing operation wx = urp y for the broken trajectories we immediately obtain the identity

F(wx )(t) = 0

for tI > 1,

that is, without loss of generality,

u(t) = expu e(t),

t-p-1

v(t) = exp v ((t) ,

t p 1 ,

Therefore we may continue this analysis in local coordinates at y, so that we consider the situation on Rn :

a X:

wx =

---+ Rn, DX(0) = Hessian of

G 0+ P '

F

at y ,

2.5. GLUING

87

Thus it is sufficient to estimate iiRwx)42 =

II const IIHwx..H1,2([-1,1])

liF(wx)42([_1,11) const

+ ii(--piiii1,2([0,1])) •

According to former discussions of the solutions for the gradient flow equation, we observe an exponential decrease at the critical point 0 of the form is(t)i

c(s) e -mt , t

O,

where the constant m depends merely on DX (0). The constant c depending continuously on s may be estimated uniformly due to the compactness of K, so that the inequality

i

o

-<.. const(u, y) e - raP, etc.

IIGIIL2([_ 1 ,0 ]) = f(1(t + 13)12 dt i

D

yields the completion of the proof.

Summing up, we can state that there is a lower parameter bound pK, such that, given any x E K X [pK, pc), the mappings Fx : Hx --+ L x and Gx : L x Hx , R(G x )= LxJsatisfy the assumptions for the fundamental Lemma 2.52 and guarantee the unique existence of a correction term 7(x) E B,(0)

n

complying with Fx (7(x)) = 0

and 117(X)111, x < 2 CK,i

and thus

117(X)iii, x < c(K)e- m° . Therefore it remains only to verify the smoothness of this correction section -y.

Lemma 2.54 The unique section -y : K x [pK, oc) ---+ F o y = 0 is smooth.

n B,(0) solving

Proof. Since we have already solved the existence and uniqueness problem, we are now able to regard the same zero point problem in local coordinates, that is with respect to a local trivialization of L- 1- and L 2 . Then we immediately obtain the assertion from the implicit function theorem together with

CHAPTER 2. THE TRAJECTORY SPACES

88

the smoothness of F. In particular, we are provided with a uniform estimate for the derivatives

I ay (x )I I

c o nst(K ) • 11 7 ( x )11

for all xEK x [PK, co) 0

Piecing together all these results we obtain the trajectory gluing as asserted above in the shape of a smooth mapping

#:

K

X [PK, CX) )

(2.79)

A 4 X,Z

-4

. exP okp, 'Y (x ) •

X

The next step is to derive the embedding property for fixed parameter p .,.. pK from the embedding property of the pre-gluing map rp and the exponential convergence iiD7(X)ii, 117(X)ii .., const e—rnP .

We shall prove this property in the following section for the gluing operation induced for the unparametrized trajectories

(i.i,f)) E M x X M y,2 . ,y

The Embedding Result and the Convergence In order to arrive at the statement of Theorem 3 we are finally going to prove the embedding property and compatibility with complementary weak convergence toward the original broken trajectory as regards the spaces of unparametrized trajectories. At first, we define thegluing operation which is induced on the unparametrized trajectory spaces M x,y • Definition 2.55 Let us consider the diffeomorphism already analysed above, A 4

xR

0: (71,

t)

Ivix,y I—'

'

at

5

for which we implicitly use the identification (2.80)

Aqx t y = •A4c;,y3

41 0

11

with respect to a fixed regular value a from (f (y), 1(x)). By reverse argumentation we can also consider the projection map

[']: Mx.v —4 A,v ,

2.5. GLUING

89

so that it holds [i10 1 = û together with f (û(0)) = a. Given a compact set of unpaiumetrized broken trajectories

R cm r ,,

X My,z

compact ,

we define the induced gluing operation on this set by 14,9 1).

= [11 #pfd =

pi) ) , (4

where r(ii,f.), p) is determined uniquely via the condition f ((fi# p f,)(7- (i , p))) =a .

This means that, after the gluing of the parametrized trajectories (flo, 0o) E M x ,y x M, we have to execute a gauge-shifting or reparametrization by x(...), in order to get an unparametrized trajectory p) E .A4 —s , z regarding the above identification. Now, we are able to prove the embedding result in version of Proposition 2.56 There is a lower parameter bound pk, such that the unparametrized gluing map :

kx

(pk-,00)

is a smooth embedding. Proof. The differentiability property follows immediately from the respective result for the gluing map #. Hence we have to show (a) the regularity, that is the property that

D#01 , f),

is an isomorphism for each triple (fi,

and (b) the injectivity of the gluing map.

(a): Due to the compactness of k it is sufficient to find a lower parameter bound for each broken trajectory i)) E k in such a way that the regularity of p) holds for all p> p(û, :6). Since the regularity of the differential of a continuously differentiable map defined upon a finite-dimensional manifold is a so-called open condition, 16 this lower bound can be found as an uppersemicontinuous 17 function on k, so that p(k) = max(Ü ) E p(i f.)) exists. 161n other words, each regular point (4,6, p) E ie X [0, oc>) with regard to the differential Di has a neighbourhood in if x [0, oo) consisting of regular points. 17p upper-semicontinuous means: x n x limp(xn) > p(x)

90

CHAPTER 2. THE TRAJECTORY SPACES

Hence, we shall prove the existence of such a bound for any arbitrarily chosen fixed pair 0. Due to the identity dim

(k x (p oo)) = dim .iax,z

0. This it is sufficient to verify the injectivity of Dîf (71, p) for all p> will be accomplished indirectly. Let us henceforth use the shorter notation = u referring to the above identification M = M. From the analysis of the unparametrized trajectory spaces we obtain the decomposition of the tangent space

a

T ^ .A/G z = fi#,of, • u#,f,

(2.81)

--

at

x•

P

at

E)171x ,z (221°)

C Ms,z .

This follows from the construction of the induced unpasametrized gluing operation with —

(ii. #130 =

at

at

T u'v , P

f (14 f)(T)) = a • P

Without loss of generality we can assume a = f (y) . differential of if — by means of (2.81) as

(2.82)

D#(11, 75 , t) = Proj T --1.,. (D#p (u,y) • (CO

Then we compute the

D#p (u, y) • (U,

• t) .

Thus, assuming that DiTt(it, p) is not injective, we find non-vanishing tangent vectors (C(,t) 0 satisfying

D# p (u, v)

() + D#(u, v) •

(it,

eR

•t

Without loss of generality we can carry out the normalization

+ ii(ii 2 +t2 =1

.

We now accomplish the proof by contradiction. Let us assume the existence of sequences pn

00,

((en, (n )) nerti bounded

and

(tOnEN C

R

satisfying the identity

(2.83)

D# p„

(n) D# p, •

(a, —1 )) = tn • 1(u# p„v)

2.5.

GLUING

91

for all n c N. Due to the construction of the gluing operation in the last section we compute (u, y) • (C () VI exP (in,'Y(Xv.)) • (e#p„()

+ V2 exp(...) - (D-y(x) • (C ()) ,

so that the sequence ( 1 D#,. H op )nEN C R turns out to be bounded. Since the sequence 1(u#,„„.v), (p) is also bounded uniformly as we see from the equation

11.4(u# P- 4, (P-)11L2 = 111(u#puv)11 L 2 = f(x) — f(z) , we deduce the same for the sequence (I \-n)nEN. A detailed calculation involving the formulas

D# p , • (Gt (T) + D#(i, —'ù) Vi exP (in, 'Y(Xn)) • (n#P,,(n

it#P,‘ (—

)) )

+ [V2 exP (Xn, -Y(Xn)) 0 D-Y(Xn)] •( • • • ) and

a =

=

a

exPx„ -y(xn))

Vi exP (Xn,'"Y(Xn)) • in + V2 exp (in, -r(Xn)) • VrY(Xn)

yields the identity (2.83) as

(2.84)

V1 exP

tna(U#%11))

(Xn,7(Xn)) ' (61# pr,(n fi#

+ V 2 exP(—) • (D'Y(Xn) • (61

— + tnVCY(Xn)) =

O

.

Exploiting the well-known identityls

(V2 exp -1 oVi exP)(1), 0) = idTM and the exponential decrease

b(Xn)ii, liD'Y(Xn)11

V,

which was analysed above in the proof of Lemma 2.54, we are led to the decisive convergence

(2 .85) 18See

n#19 Xn

Appendix

A.2.

Yn

p— + tn A(u#(;)n y)11 1,2 —40 ti#„,(0

.

92

CHAPTER 2. THE TRAJECTORY SPACES

According to the definition of the linear gluing version and of the pre-gluing operation, this implies convergence toward zero

11xn +

( 1 + tn)II

IlYn + 1, (tn

=0

1)11H1.2([1))

,

that is

(2.86)

Xn (1 + tn) u —■ 0 and yn + (t, — 1) • v — ■ 0 .

Taking into account that, due to the assumption and the decomposition (2.81), the relations E yn 6

Tu.Aqx , y , TSI x , y n Rt = {0}, TvA,z, Tv.kiy,z n Ri) =

{

o}

hold and that Tn.A4 x ,i, and T,M 1,, 2 are finite-dimensional, and thus locally compact, we observe that the zero-convergence in (2.86) leads to and

xn —■ 0

tn —■ —1

and y,. —*O and

+1 ,

tn

respectively. Hence, we deduce a contradiction. (b): We now have to verify the injectivity of now assume the existence of sequences Pn —0 co

and

:

kx

[pk,00) --+ M. Let us

fh,n) (112,n,

E

k,

which satisfy the identity [fi 1 ,n#17, 1)1,n] = 2,n#,90)2,n)

for all n E N ,

where we refer to Definition 2.55, that is, = (fii,n #pn i)i ,n ) ri„ , i = 1,2

with f ((...),,,„ (0)) = a. Resorting again to the compactness of consider without loss of generality the convergent sequences

i = 1,2 .

fii,n

Then the representation exp

'

Y(Xi.n), Xi.n =

f/i,n

k

we can

2.5. GLUING

93

together with the exponential decrease

7(Xz,n)

fl

00

0

yields the following H1,2 -convergence with respect to local coordinates:

(2.87)

o

11(111 #. 1)1 ) -ri,

(1123n,‘ 1)2 )r2, ,, 111.2

°•

According to the construction of the pre-gluing map #0 , the fixing of the time-parametrization

f (hi (0)) = f (112(0)) gives rise to the asymptotic synchronization 71,n ——)•

and thus to the identities ûi

= ft2 = û

and

4)1 = '62 = 47) •

However, knowing from part (a) that At — is a local diffeomorphism, in particular at f.2) E k, we find an initial index no E N, such that the identity (fil,n,f) 1,n) = (fL2,n., 02,n)

holds for all n

no in contradiction to the assumption.

At the end of this analysis of the gluing operation for unparametrized trajectories of time-independent gradient flow, we now discuss the relation with weak convergence toward broken trajectories as was defined in the last chapter.

Proposition 2.57 Given a broken trajectory (fL, i:)) from x . 1V --11) , z , each arbitrary increasing sequence of parameters pn gluing opthe oc together with eration for unparametrized trajectories # induces the geometrical convergence19 fi# pj)

(IL

^

V)

The converse is also true: Any sequence of unparametrized trajectories converging to a simply broken trajectory finally lies within the range of such a gluing map #.

Proof. The CT -convergence toward ft and arises from a respectively appropriate choice of the value a with regard to the identification .A4,,. = Firstly, let us choose a value satisfying 1(z)
T0

of reparametrization

CHAPTER 2. THE TRAJECTORY SPACES

94 that is

((,1 4 p ,‘ f, ) (0 )) 7i EN

c ma

.

Due to the compactness of Ma we can find without loss of generality a limit point xo E Ma,

fiîi p„ ;6(0)

SO

•

Considering the unique trajectory s E C°3 (R, M) satisfying

= —V f 0 s

and

s(0) = xo

we are led to the weak convergence

s

û p j

where the reparametrization sequence rn for

(7.1# pa t)).„ =

[7(Xn)1-r,

is determined by f ((f4p,)(rn)) = a •

The exponential decrease of -y(xn ) analysed above implies the estimate

ly(x,i )(rn ) I

const • e —'"

0

and thus the convergence

(ii#p°, f;)(7-7,)

xo .

Hence, due to the construction of the pre-gluing operation, the condition f(x0) < 1(y) gives rise to the limit point

lim f/(r„ —

n

that is,

=

/JO

= xo ,

j.

An entirely analogous argument now starting from the assumption f(y) < a < f(x) yields weak convergence to the other part of the broken trajectory, coo

The converse statement is an immediate result from the uniqueness provided by the Banach fixed point principle by means of which we developed the gluing maps. If we are given any sequence of trajectories with fixed endpoints which converge weakly toward a specified simply broken trajectory, the elements finally lie within a H1,2 -neighbourhood in which the above contraction mapping principle (see Lemma 2.52) works. Hence, these elements have to lie within the range of the gluing map belonging to this specified simply broken trajectory. 111

Thus, we have accomplished the proof of the central gluing Theorem 3.

2.5. GLUING

95

Conclusions At the end of this section, regarding the application within our homology theory in question, we wish to discuss some consequences of the compactness-gluingcomplementarity which has been proven now. Actually, we shall consider the unparametrized trajectory spaces of second order and the simply broken trajectories. This means that we restrict ourselves to the relative Morse index p(x) — = 2. Then we are able to classify the connected components of the one-dimensional manifold without boundary .A4 x ,, by the diffeomorphism type of Sl and (-1,1), respectively. We obtain the following

(-1,1) be a diffeomorphism. Then there are Corollary 2.58 Let ct, : exactly two different broken trajectories ('12, j) 2),

X Mz; i =- 1,2

ITO E

satisfying

gx ) — 1 = /1(Yi) = ,a(y2) = 12(z) + 1 , and, given any small e > 0, we find gluing parameters pi (E), p2(€) E IR, such that we obtain the identities

( 0 # ) ({fil x }

x

(P1, 00) -= ( -1 , —1 e)

and (0 0 #)(1 11 21 x If)21 x (p2, Do)) = (1 —

E, 1) .

Proof. The boundary property is a consequence of the gluing theorem. The fact that the broken trajectories cannot be identical arises from the uniqueness of the correction term -y(fli epf4) with respect to the uniquely supplied bundle LI with regard to large gluing parameters. This is the same argument which was used for the last statement of Proposition 2.57. From this corollary we gain an equivalence relation for simply broken trajectories with relative Morse index 2. This equivalence relation will prove crucial for the construction of Morse homology. In fact, it is the kernel of what we called in the introduction the cobordism relation for trajectory spaces.

Definition 2.59 Let us define the set of simply broken trajectories with fixed endpoints .A4 z ,z = { (fi, ) E .A4 z , y x .A4 y,z I A(x) — 1 = ii(y) = /./(z) + 11.

Theil_the gluing cobordism from Corollary 2.58 induces an equivalence relation on M x , z by , f)i)

(û2 ,i 2

)

)

(112#,, 2 i)2)

in M.,.

•

CHAPTER 2.

96

THE TRAJECTORY SPACES

This means that these respective broken trajectories correspond to the ends of the same pathwise connected component of M x , z , so that they are cobordant in the sense of 'compactness' and 'gluing'. It is obvious from the possible diffeomorphism types of one-dimensional manifolds without boundary that it holds that = 2 for all (û,i) E .A4x,z • We can thus decompose the set M x , z into pairs of different equivalent broken trajectories. -

2.5.2

Gluing of Trajectories of the Time-Dependent Gradient Flow

The above defined equivalence relation for simply broken trajectories of the time-independent gradient flow yields the crucial argument in the proof of the chain complex property 82 = O. We now require the corresponding gluing results for the time-dependent and the .X-parametrized situation, in order to prove the homotopy invariance of the Morse homology in an analogous way. The essential difference between the earlier gluing operation and the gluing operation which is to be constructed later, is a lack of invariance under timeshifting

U • T = U, with regard to the actual homotopy trajectories. However, we notice that we merely admitted Morse homotopies for the time-dependent gradient fields which were asymptotically time-independent, that is,

8

- (Vht) = 0 at

for

R.

Therefore, we must distinguish three different types of gluing operations: • gluing of shifting-invariant trajectories, as analysed above, • gluing of broken trajectories, which consist of shifting-invariant pieces as well as of trajectory pieces corresponding to the time-dependent gradient field, called mixed broken trajectories, and • gluing of broken trajectories which do not contain any shifting-invariant pieces at all. We shall obtain the result for the mixed broken trajectories analogously to the first gluing theorem. Let ht be any regular Morse homotopy .P3 •

2.5.

GLUING

97

Theorem 4 Given a compact set of mixed simply broken trajectories AAh

x .tilf° xp,yfi

kc

there is a lower parameter bound p- and a smooth embed-

ding

k

#

x [p , oo)

M xh .,ya U h#P ii

(uh,i),P)

such that, given any arbitrary sequence of gluing parameters pn -> oc, we are provided with the weak convergence

(un , f))

uh#p„v

.

Proof. Essentially, the construction of this gluing operation for mixed broken trajectories differs from that in Theorem 3 merely by the definition of the pregluing map. Instead of shifting both parts of the trajectories by the value p in opposite directions regarding the time-parametrization, we now define the pre-gluing map #° for the mixed broken trajectory h

t3 (uh, y) E M x,k ,zo X M x13,Y0

by t p - 1 (2.88)

p[eXP; (Uh)]

(uhepv)(t) = expx

(t), It

+p /31-fexpx-(v))-2p 01

t

(t),

pl

1

p+1.

In this framework, the lower bound po for the gluing parameters is now additionally bounded below via the time interval during which the Morse homotopy f' is active, that is po ?- R(ht).

The construction of a smooth gluing operation for these trajectories is accomplished by steps entirely analogous to that of the former gluing operation. What proves decisive for transferring the essential stages of the construction, is the invariance under time-shifting of the HI-2 - and L2 -products along the trajectories. It is worth mentioning, that this time we can consider equivalently the gluing operation as an embedding #p K .A4 h

of compact sets K C x .A4 z as well as the embedding stated above, #

x

co)

for fixed gluing parameters P ?-PK

.A4

•

98

CHAPTER 2.

THE TRAJECTORY SPACES

Once again, it is immediate from the representation Uh#p,,IJ = eXp u h#Lf, 7(Xn)

const e— QP- —> 0, that the original trajectory together with lry(xn )(t), (uh, 0) may be reproduced from the weakly convergent sequence uh

# ,9 3

(vi, 4)) •

0 The respective result for mixed broken trajectories of the form Vh) :ICL,Yot

X it41.Y1a,YO

is obtained likewise. Broken Pure

h-Trajectories

For the case of the gluing of broken pure homotopy trajectories, that is a similar operation which glues trajectories (uo, uo-r ) E ./Vi hx: 13zo X Mxh:7x, and maps them into the trajectory space Jythz 7 we should present a separate approach. The crucial feature in this situation is that the gluing parameter p, for which the above gluing construction gives rise to an embedding into a trajectory space, if p is large enough, now also determines the target space M zh :rx , itself. The result is stated as Proposition 2.60 Given a compact set of broken h-trajectories

K c M hc° h'31 x 0 ,xox M xp,x, there is a lower parameter bound Ro , such that for any R Ro exists,

f

(R)

a homotopy

f" ,

together with a smooth mapping

#R K which represents an embedding. Proof. For this case of the broken trajectories original, 'symmetric' pre-gluing:

(uao,

uo-i ) we use again the t —1

(2.89)

(ueR v)(t)

lexp; 1 Uoi31 R

= e/ u:P3r( : + 4 (+ 0+ [eXPx : 110.7]— R) vp-i (t —

t I.)

It '.5 1 t _.›.- 1 .

2.5.

GLUING

99

In order to obtain the correction term for an h-trajectory by means of a contraction map, we choose a suitable homotopy between f and P3 , namely

(2.90)

h7i1 (t) =

{ h' 13 (t -F R, - ), t hf3"Y(t — R, .), t

0

0

Provided that the gluing parameters are large enough, R ,.. Ro, this homotopy is well-defined. We are now able to construct the required gluing operation #R using the same methods of Banach calculus as above. Essentially, the difference between this gluing operation and the gluing in the first theorem of this chapter consists of the fact that the invariance under time-shifting for the trajectories of a given time-independent gradient field has now been replaced by the appropriate choice of the time-dependent gradient field in the case of explicit time-dependence on both trajectories. 0 Concerning this type of gluing operation we wish to discuss a special case as given in the following Corollary 2.61 Given the situation of isolated h-trajectories, that is ii(xa) = ii(x0) =

there is an R> 0 such that the associated gluing map #R : Arl zh :3,za x is a one-to-one correspondence of finite sets.

Proof. According to the above result it is sufficient to verify surjectivity. Let hc-7(R) ual, be an h-trajectory from M x.,s, . Then, consideration of the solutions uo•

E M h"r0

and

uf3.7

ho, E Mso,x_r

of the respective time-dependent gradient flow, which are determined uniquely by Ti c,o(0) = ua.1,(—R) and u(0) = leads us immediately to the broken trajectory forming the pre-image with respect to #R. El

2.5.3

Gluing for A-Parametrized Trajectories

Let us now consider the manifold MI/ ,ys c [0,1] x pL,2,10 of A-parametrized trajectories as a zero set with respect to the mapping G'S : [0,1] x

—t L 2 (pL,2 va . Tm)

1

(A, u) 1--* ti + — VH°(A, • ) o u . ■./

CHAPTER 2.

100

THE TRAJECTORY SPACES

H:5

Here, HO describes a A-homotopy h o hl of regular Morse homotopies between fa and f 0 . In this framework, too, we wish to discuss the problem of a gluing operation combining such A-parametrized trajectories (A, u) with shifting-invariant trajectories of the gradient flow with respect to fa and f 13 , respectively, such that we again obtain A-trajectories (A', y) with A' generally differing from A. Theorem 5 Let k G x ma z, be a compact set of mixed simply broken, A-parametrized trajectories (A, UA, f)). Then there is a lower parameter bound pie and a smooth embedding

# 11 (A,70„ , p)

(A, uA)# pil f) =(itw-5,) ,

such that, provided any increasing sequence of parameters po co, we obtain weak convergence of the form

w5,(A,u)„ii , po )±). c °` (u)„/6)

and

A-- (A,u)„ii, po )

A.

As to the underlying analysis, we do not come upon anything new. We merely expand the Banach manifold, within which we are searching for the zeroes of a Fredholm map by means of the contraction principle, by the compact interval [0,1] of A-parameters. Therefore, we will not provide a detailed proof at this stage. The strategy for proving this theorem is organized in two stages in a way which is absolutely analogous to the above investigations: At first, we have to define an appropriate pre-gluing map, in this situation similar to the timedependent case in Section 2.5.2, together with a suitable normal bundle LI, which now lies within R X Finally, we must derive a unique (P' 2 smooth correction section within this normal bundle by means of a suitable contraction map.

42

Definition 2.62 Referring to the corresponding pre-gluing map for mixed broken trajectories from Section 2.5.2 for an orientation, we now define the pregluing operation #11'' : K x [po, oc) [0,11 x P1],.2, zo by (2.91)

(A, u),) #pH ''v = (A, uAirp v) ,

where we denote by ep the operation from (2.88), that is

(uxep v)(t) =

tp—1 u(t), v _2p (t), t p +1

, etc.

2.5. GLUING

101

In strict analogy with Definitions 2.48 and 2.49 we find a construction of a linear version of a pre-gluing map #x , this time for x (A , ,u), )#pH,o v E

Ps12 zo . Thus, we obtain a normal bundle Li- from

[0,1]X

Proposition 2.63 There is a lower parameter bound pi ?- po such that the -4 Fredholm operator DG'0 0t,u),epv) : R x onto for all gluing parameters p pi and mixed broken trajectories ((A, to,), E k. Additionally, the composition of the linear version #x with the (R x L 2 )projection onto ker DGa0 (x) then induces an isomorphism:

PrOiRkext L1,03( x ) #x ker DGa0 (A, 10,) x ker Dt,

ker DO:0 (A, u), rpv) .

As to the proof. Essentially, the proof of 2.50 remains the same up to expansion by R. At this stage we wish to point to the trivialized version, which shall be mentioned again in the next chapter in relation to the discussion of induced coherent orientations. This will be treated in Proposition 3.14. Regarding a detailed proof in the trivial case together with the additional set of real parameters, the reader is also referred to [F-1-1]. This linear version of gluing in the trivial case is of the type R H 1,2

x H 1,2 3

0=p . 4_ /44 e . (-2p) ER x H',2 .

,_,

D Finally, considering the normal bundle L I over k x [pi, oo), which is induced by this proposition in analogy with the former bundle and given a large enough lower parameter bound p2 pi , we are again able to find a unique smooth correction section of

(2.92)

: k x [p 2 , 00 ) -y(A, u)„

LI

p) = (r, ) E R x Hui

,

such that (A, u),) # pH y

+

T, exp

#,t,

E

AAH

represents the required gluing operation. Once again we observe exponential convergence (7-, e) (0,0) as the gluing parameter tends to infinity, p —4 00 .

Chapter 3

Orientation Summing up the analytical foundational results we have developed up to this stage, we notice that this knowledge about the trajectory spaces of the timeindependent and time-dependent negative gradient flow enables us already to build a Morse homology theory with coefficients in the field Z2. However, in order to admit arbitrary coefficient groups, i.e. coefficients in Z, we still have to accomplish more elaborate results concerning the characteristic intersection numbers for the unparametrized trajectories. Referring to the introduction, we may deduce these intersection numbers from a comparison of the canonical orientation of the intersection manifold

Wu (x) ch W s (Y)

Mx,v

by the negative gradient field with some coherent orientation related to the critical points. Considering the framework of our analytical methods we need a concept for such a coherent orientation of the trajectory spaces, which are treated as zero sets of the fundamental Fredholm map with respective endpoints. Actually, within this analytical context, coherence denotes the compatibility of the respective orientations with the cobordism relations from the compactnessgluing-complementarity in the last chapter. In fact, the purpose of this chapter is to extend the former cobordism concept for trajectory spaces to a concept of oriented cobordisms. The fundamental feature of this concept will be the notion of orientations for Fredholm operators arising from the already well-known determinant bundle (see for instance [Donn. The reason for using this determinant bundle for Fredholm operators may be explained as a generalization of the orientation of a manifold. Knowing that in our framework the tangent space of a trajectory manifold can be identified with the kernel of the surjective linearization of the fundamental Fredholm map, we wish to drop this regular-

104

CHAPTER 3.

ORIENTATION

ity assumption and define a substitute for the maximal exterior product of the tangent space, in which a non-vanishing element represents an orientation. This substitute concerning a non-surjective Fredholm operator consists of an appropriate combination of kernel and cokernel, so that a continuous variation of a thus oriented Fredholm operator preserves the orientation, whether the operator is onto or not. Hence we shall be provided with an orientation concept for trajectory spaces including the regular trajectory manifolds as special cases. Another point which seems worth mentioning concerns the topology of the underlying manifold M in the sense of the axiomatic homology theory in question. It appears to be founded in an abstract way within the existence and construction problem of such a coherent orientation for the Fredholm operators Du E E.*Tm as u describes the trajectories of the negative gradient flow. Throughout the development of this analytical concept of a coherent orientation we keep in close analogy to the Floer homology in the symplectic case. The corresponding results can be found in [F 11]. At first, we shall review the construction of an orientation bundle for Fredholm operators in general. Then we shall verify the compatibility with the gluing calculus for Fredholm operators of the type F E Etriv . In the second half of the chapter we shall transfer the orientation concept from the trivialized framework to the manifold M, so that the topology enters when we ask for the existence of a coherent orientation. In contrast to the symplectic case in [F-HI we shall also consider non-orientable manifolds, whereby we shall need a slightly modified concept. -

Finally, we still have to verify that this method for obtaining a coherent orientation on the trajectory manifolds based on the Fredholm calculus is equivalent to the classical process of induced orientations from the differential topological viewpoint. This shall be postponed to Chapter B of the appendix.

3.1

Orientation and Gluing in the Trivial Case

3.1.1

The Determinant Bundle

In order to define the orientation bundle for Fredholm operators, we require the following preparations: Definition 3.1 Let E and F be finite-dimensional R-vector spaces. Then, by means of the short notations for the one-dimensional vector spaces

AE = A dim EE and A ° E = R ,

3.1. ORIENTATION AND

GLUING IN THE TRIVIAL CASE

105

we define the one-dimensional space of determinants with respect to E and F by Det (E, F) = (Am axE) E) 0 (Am

F)4

Now let Ho and H1 be fized Banach spaces and let us denote by

.F Ho H i ) C C(110, HI) (

,

the open subset of Fredholm operators. Then we define the space of determinants associated to such a Fredholm operator F E .F(Ho, H1) as Det(F) = Det(ker F, coker F) .

If we further consider the continuous mapping of an arbitrary topological space X into the space of Fredholm operators

f : X 4 .F(Ho, Hi) , —

we may define the space Det f U {x} x Det(f(x)) . x EX Thus, the canonical projection map is endowed with the one-dimensional Rvector spaces Det (x)) = (x) as fibres. The aim of this section is to show that the space Det f is a real line bundle on the topological space X. Then, the local trivializations uniquely determine the topology of this vector bundle. If we assumed dim ker f or dim coker f to be locally constant functions on X, 1 this result could be immediately concluded. By the following fundamental algebraic lemma we will derive the existence of the determinant bundle for general Fredholm operators without this assump-

tion. Lemma 3.2 Let 0 -4 E1

di

. . . dj-1V

be a sequence of finite-imensional 1W-vector spaces. Then there is a canonical isomorphism (Am"Ei ) 2-4 e) (A"Ei ) .

:

i even

i odd

1Note that due to the continuity of the integer valued Fredholm index the continuity of one of the two functions implies that of the other.

CHAPTER 3. ORIENTATION

106

Proof. Let eii, , eln, be a basis of El. Since dl is injective we may extend the linear independent vectors dl (ell), - • • , di (ein, ) by e21 , ,e2n2 to a basis and due to exactness, we can find E2. Corresponding to this first step of successively at each stage of the sequence an extension to a basis of the form for i

di(eii), • • • 7 di(ei707 ei+i 1, • • ei-f-i

1,

,

k —1 ,

with nk = 0. Then we define the isomorphism 0 by d;(eii) A

...

0d3(en)

A

A di (ei ni ) A e21 A... A e2n 2 • • • A e2i 112j .•. A e4n4

•••

en) A ... A d2(e2 2 ) • • • A eini Aen 3 0 • . - 0 • • • A e23 ,-i-1 nr2 2 ,-F1 A ... A e3n (®d2(

en A

Obviously, this isomorphism is independent of the special choices of the vectors of the bases, because any change in the bases operates by multiplication with the same determinant of the transformation on either side of the homomorphism

0.

0

This lemma enables us to provide local trivializations of the determinant bundle 7r : Det f —, X. (See the careful exposition in [F-HI for details.) Given x E X, we associate to the Fredholm operator f(s) a linear map .0 : Rn -- H 1 , such that

(3. 1)

4,(x) :

Er

X HO —+ /11

(h, k)

I—) 1,G(h) + f (x) - k

describes a surjective Fredholm operator. Since surjectivity is a regular property, it must be satisfied on an open set in X. Thus there is a neighbourhood U(x) such that the surjectivity of To(Y) E

T(Rn X

1107 11 1)

holds throughout U(s) together with the identity ind

(y) = ind I:0(x) for all y EU(S) .

Thus, the mapping

ftP(Y) Rt n X lio (h, k)

—+ Rn X Hi (0, h,(y)(h, k))

gives rise to a real line bundle Det fo on U(x), because we now deal with the constant functions dim ker f,/, (y) and

dim coker Ai, (y)

3.1. ORIENTATION AND GLUING IN THE TRIVIAL CASE

107

on U(s). By the fundamental Lemma 3.2, we can verify that these local determinant bundles associated to the covering

U U(x)=X

xEX

yield a local trivialization of Det f on X. Actually, given (x0/;,U(x)) and y E U(s), we have an exact sequence 0

ker f(y)

ker fo(y)

d2—n

d3 coker f (y)

0,

where

di (k) = (0, k)

d2 (h, k) = h d3(h)= This induces a natural isomorphism Det f(y)

(Am"ker ftp(y))

(Am"Rn ) *

,

i.e. a natural isomorphism

Det f(y)

(3.2)

Det f,p(y)

due to the surjectivity of "; (y). The corresponding calculation of the transition maps therefore provides the result that 7T :

Det(f)

X

is a real line bundle. Definition 3.3 The bundle Det f is called the determinant bundle of f and

any non-vanishing section in this line bundle induces an orientation for the family f: X -4 1(110, Hi)

of Fredholm operators. We remark that such a section does not necessarily exist.

3.1.2

Gluing and Orientation for Fredholm Operators

Throughout this section, we shall use the results of Fredholm theory developed in the last chapter. In order to define a gluing operation for Fredholm operators of the type' FA 2See Definition 2.14.

E Ee

Etriv

= Tr" Rn

CHAPTER 3. ORIENTATION

108

analogous to the construction in the last chapter, we must restrict ourselves to asymptotically constant operators, that is, to K = AK E Etriv satisfying 'WO = const

T for a T O.

as It'

Actually, referring to the equivalence relation KA LB •#• =

B1

and recalling that according to Lemma 2.15 these equivalence classes are topologically contractible, we may choose an asymptotically constant representative from each equivalence class eic. Definition 3.4 Considering the isometric R-action

E

RxE

(p,FA )

1- 0

Fp Fdie

p)

we are able to define the gluing operation for asymptotically constant operators K,L E E with matching ends K+ = L- by

K# pL

E e(K - , L+)

=

{ AK (t + p), t 0 A L (t - p), t0

A(t) =

for all p po(K,L). Here, the lower parameter bound is determined by the asymptotical behaviour AK(. + P)i FL*

E

4 A

AL(.

- p)1

for all

p

po

•

Remark According to the index theorem from Section 2.2, the Fredholm index

operates additively under this gluing operation, i.e. ind (K# p L) = ind K + ind L holds for all asymptotically constant operators K,L E E with matching ends K+ = L - . Moreover, the equivalence class OK#L, does not depend on the actual gluing parameter p and the respective choice of the representatives K E OK, L E AL. In a nutshell, if we consider the set = E/ of equivalence classes [K] = OK K E E and the map ind :

z,

3.1.

ORIENTATION AND

GLUING IN THE TRIVIAL CASE

109

we can state the following relation:

ind o # = + o (ind, ind ) . The crucial question now is whether there is a similar orientation concept for Fredholm operators as defined above, which is likewise naturally compatible with the gluing operation. At this stage it is important to state that, due to the contractibility of the equivalence classes [K] = 43(K+, K- ), the associated determinant bundles

Det [K] -■ 0(K+ , K - ) are in fact orientable. First, we require an isomorphism between the spaces of determinants associated to the respective Fredholm operators

Det(K) 0 Det (L)

Det (K# p L) .

Second, regarding the equivalence classes OK, OL and OK# p L, we have to verify that the orientation on eic#p L induced by this isomorphism is independent of the choice of the actual representatives. Once again, the fundamental Lemma 3.2, by which we may transform in a natural way the Fredholm operators K and L into the surjective operators kv, and L,p, turns out to be the decisive utility. Thus let K and L be asymptotically constant operators from

Etriv

C G (H 1 '2 (R, IV),

L2 (R, Rn))

with matching ends K+ = L- . We find a linear mapping lp : Rk --,

L2 (R, IV ) ,

such that the Fredholm operators

K,,1,, L,,1,: R k x H 1,2 -■ L 2 (h, u)

1-0 L - u + 71, • h

(h, u)

i- K - u + II; - h

are both onto. By analogy with the derivation of the determinant bundle, Lemma 3.2 supplies us with natural isomorphisms

Det(K,p) -= ■ Det(K)

and

Det(L,p)

Det(L)

concerning the operators Kv„ Lo : Rk x H 1 '2 --■ R k X L2 (a, u)

(a, u)

1-4 (0, L • u + il, • a) 1-■ (0, K • u + ti) • a) .

110

CHAPTER

3. ORIENTATION R

We note that without loss of generality we may assume an

supp(0(a)) C [—R, R]

for all a E

0 satisfying

,

because the bounded linear mapping

OR

L2

L2

U 1—■ OR • u associated to a smooth cut-off function fiR with compact support in [—R, gives rise to a continuous deformation satisfying

LoR4

c(

p.,2 ,L 2 )

L IG as R

oc ,

and the surjective Fredholm operators form an open subset of the space of operators £(H 1 '2 , L 2 ). Given these surjectively extended Fredholm operators ko and L„ we now define a gluing operation which is adapted to the situation:

Definition 3.5 Let po = po(K,L) be the lower parameter bound associated to

the asymptotically constant operators K and L. In order to take into consideration the extension by W, we lift this lower bound to pl = R +1 + po, and we consequently define the mapping

ko# p 4 - , : Rk x Rk x H 1,2

L2

for P Pi by

(k

p i, )(ab, u)(t)

((K # p L) u) (t) + W(a)(t +

+ 7P(b)(t — P)

The central proposition with respect to this gluing of surjective Fredholm operators may be stated now as

Proposition 3.6 Let P,, be the orthogonal projection in the Hilbert space

Rk x

k x L2

onto the finite-dimensional subspace ker (k p i-o). Then there is a lower parameter bound p2 ?...• pi such that the following holds for all gluing parameters p p2 : The Fredholm operator ping

k v,#p -L,,b is surjective and the linear map-

: ker k,, x ker ((a, u), (b, v)) is an isomorphism.

ker (k#:4) 1— ■ Pp (a,b, up + v_ p )

3.1. OPIENTATION AND GLUING IN THE TRIVIAL CASE

111

Proof. We observe that this proposition corresponds to Proposition 2.50 about the linear version of the pre-gluing of trajectories up to the finite-dimensional surjectivity extensions (ilk ,1p). Regarding these extensions with compact support supp 11)(a) C [—R, R], we may accomplish the proof by strict analogy. The reader is also referred to [F-H]. Moreover, this isurjectivation' by means of (I .. , 0) comes to the core of the discussion of the trajectory gluing for the A-parametrized gradient flow with respect to the orientation problem. There, too, the Fredholm operators Du become surjective generally at the earliest after extension by the partial derivative for the variational parameter A. 3 Now considering the induced mapping :

x istk x H 1 ,2 _4 Rk x Rk x L2

(Ko # p L o )(a,b, u) = (0, 0, (ko # p L ip)(a, b, u)) , which is equivalent to

K# p L,p

,

according to the above derivation of the determinant bundle, we are provided with the natural ismomorphism (3.3)

Det(K o #pL o )

Det(K# p L)

by Lemma 3.2. Noting that the gluing Proposition 3.6 induces an isomorphism (Am"ker kip) 0 (Am"ker L ip)

A'ker (koif p L ip)

and resorting to the natural isomorphism (AmaxRk)* (AmaxRk)*

(Amax (Rk X Rk )) *

we obtain an isomorphism for the associated determinant spaces of (3.4)

Det(K,) Det(L ik )

Det(Kv,# p L,p) ,

provided that the gluing parameter p is large enough. Finally, the natural isomorphisms yielded by the algebraic Lemma 3.2, which describe the transformation of the determinant spaces with respect to the surjective extension, i.e. Det(K0 ) Det K, etc., lead us to the isomorphism for the original determinant spaces (3.5)

Det K

Det L

3See also Theorem 2 in Section 2.3.2.

Det(K# pL) .

112

CHAPTER 3.

ORIENTATION

The next step is to verify that this isomorphism may be transferred to the determinant bundles of the associated equivalence classes ex, 0/, and ex#L• In other words, we have to verify the coherence with respect to different representatives. Since the equivalence classes as topological spaces are contractible within Etriv , , we may choose connecting arcs K' and Li with T E [0,1] for each pair of representatives (K°, K') and (L°, L 1 ), respectively. We are able to repeat the above analysis starting from these arcs instead of fixed operators together with the additional argument that the set of parameters [0, 1] is compact and the arcs are continuous. Therefore, we once again find surjective extensions and corresponding isomorphisms uniformly with respect to T or dependent on T. These isomorphisms give rise to a vector bundle isomorphism on [0, 1], i.e.

Det KT

(ICT#AT)

Det LT

(3.6)

[0, 1]

for p ?. po(K [° ,1] , L1011 ). Hence we deduce that our orientation concept for the equivalence classes OK and fiL is compatible with the gluing which yields OK#L•

Before we finish this section by stating the result in a theorem, we wish to mention an important property. As we already noticed in the section on trajectory gluing, due to our construction, this gluing operation itself does not satisfy an associativity rule. However the mere orientation problem can actually be solved. Let K, L, M E Etri, be asymptotically constant operators with matching ends K+ = L- and L+ = M. Then, provided suitable gluing parameters , , p4 , the operators (K# 1 L)# 1,2 M as well as K# 4 (L# p3 M) are well-defined representatives of the same equivalence class e(If , M+). Thus we can construct a smooth line bundle E on [0, 1] with the boundary fibers

E0 = Det((K# p, L)# p, M) and a vector bundle isomorphism

and

EI = Det (K#p4(L#p3M))

113

3.2. COHERENT ORIENTATION [0,1] x (Det K

Det L

Det M)

E

\V [0,1]

such that (Do and (D i represent exactly the compositions of the gluing isomorphisms from Proposition 3.6 in the respective order. We sketch the proof. In principle, it resorts to the same analytical construction elements which we have used in our discussion about gluing up to this stage. At first, we define a gluing operation for three operators simultaneously with two free, independent gluing parameters. Then we have to prove the associated isomorphism property with respect to the surjectively extended Fredholm operators. Secondly, we have to show that, given gluing parameters that are chosen appropriately, we are able to homotope this isomorphism for three operators to the respective composition. The crucial step consists of finding a uniform estimate together with a homotopy to 0 for the difference term between the respective isomorphisms as the gluing parameters become large enough. Finally we are able to sum up this section in the following theorem:

Theorem 6 Let K and L be asymptotically constant operators from Etri, with matching ends K+ = L- and let oK and o L be orientations of the canonical determinant bundles on eK and eL, respectively. Then for all p po(K, L), the gluing operation

K# p L induces an orientation oK#0i, on ex#„L = êK#L independently of p. This induced orientation does not depend on the actual choice of representatives K E 9K and L E Or,. Moreover, the associativity rule of is fulfilled, (oK#00#0m 0K#(01.#0m)

3.2

Coherent Orientation

The outcome of the above analysis in the case of the trivial It-vector bundle IfTt x Rn is an orientation concept which is supplied uniformly for all asymptotically constant operators with coinciding ends. This means an orientation for

114

CHAPTER 3.

ORIENTATION

the equivalence classes OK E 2, and it is compatible with the gluing operation. We now intend to transfer this result to the situation of the R-bundles u*TM induced by the gradient trajectories u. Thus we principally have to consider the dependence on the curve u. Actually, this a priori issue touches on the topology of the underlying manifold. The crucial observation will be that mainly the knowledge about asymptotical behaviour suffices and that we are able to orient all Fredholm operators uniformly as one having identical end terms and associated to curves with coinciding fixed endpoints. The only delicate point is the question if two given smooth curves u and y with identical endpoints u(±oo) = v(co) give rise to pull-back bundles u*TM and v*TM admitting trivializations coinciding appropriately at 77a( c,3 )M = T(±) M. If this is guaranteed, we may transfer the results from the last section in a unique and canonical way. It is at this stage that we come upon the problem that not all two curves with coinciding endpoints satisfy this trivialization condition on the pull-back bundles in an a priori way. The obstruction of a common trivialization in the above sense is hidden in the topology of the connection to a loop u - v -1 (provided a suitable reparametrization). Actually, the vanishing of the characteristic class wi ((u v') *TM) is the necessary and sufficient condition for the trivializability of the bundle (u • v -1 )*TM. In other words, we find simultaneous canonical orientations except for curves with coinciding endpoints, which close up to n-dimensional `Möbius bands'. This topological obstruction is in fact an expression for the orientability of the underlying manifold M. Nevertheless, a more detailed analysis will show that already the coincidence of the trivializations at one end proves sufficient, if we only can specify in a uniform way which end is the right one. Then, the possible difference at the other end, which may be expressed as a fixed reflection, turns out to be irrelevant. The only essential feature is the uniform and necessarily noncanonical specification for all curves, at which end we wish to have the identical trivialization. Although it might look unexpected or even cryptic to the reader, it is worth mentioning at this stage that this specification is in one-toone correspondence with the dimension axiom of the later homology theory and with the obstruction to the well-known Poincaré duality in the case of closed but non-orientable manifolds. We shall come back to this issue in the section 'topology and coherent orientation' within the next chapter which is central to this monograph. Finally, having accomplished the transfer of the above notion of an orientation to the non-trivial framework of curves, we shall develop the concept of a coherent orientation by means of the compatibility with the gluing operation for trajectories from the last chapter. This coherent orientation is the main feature which allows us to set up a homology theory with coefficients in Z.

3.2.

115

COHERENT ORIENTATION

3.2.1

Orientation and Gluing on the Manifold

M

We shall accomplish the orientation process for the Fredholm operators from Eu-Tm not only for the trajectories of a special gradient field but in general for all smooth compact curves u E C (IR, M). Therefore, we consider this space of curves to be equipped with the Whitney C'-topology. 4 Since we merely deal with pull-back bundles u*TM on Lemma 2.19 guarantees that the induced covariant derivation from TM is Fredholm admissible. Hence, we drop V from the notation. Definition 3.7 We call two Fredholm operators K E

Eu•T Ai,

LE EV .T M

along supporting curves u, y G C e° (rt, M) equivalent if the asymptotical identities u(+co) = v(±oo) and K ± = L± hold. Any pair of trivializations

1,bv :

u*TM

x

v*TM

x WA

is called admissible for such equivalent operators (u, K) (v, L), if the identity Ou( -00 )= 1'v(-0o) holds at the identical lower ends of the curves, the relation (

±1 1

(+o0) • Ov(1-oo) l --= 1 at the identical upper ends and if the equivalence defined above for the trivial framework triv 1,bv 1,027 1 = TIMM Ou(K) = Kql is given. Let us denote the equivalence class of (u, K) by [(u, K)] or merely by [K — , as it is already determined uniquely by the endpoint operators K 1 E End(TM). Remark Given two curves u and y, which are connected to a trivializable loop with respect to a suitable reparametrization, i.e. w1 ((u y -1 )*TM) = 0, we always find an admissible pair (0u , V),) associated to (u, K) v, L), so that 4Relating to the differentiable structure on R introduced in the first chapter, we note that weak and strong Whitney topology on CC.° M) fall together.

CHAPTER 3. ORIENTATION

116

an nthe identity 95/4(±00) = i,b,,(±oo) holds. Otherwise, if u • v dimensional 'Möbius band', we choose asymptotical trivializations C(±co) : Iftn, such that C(±co)K ± C(±oo) -1 = C(±oo)L ± C(±oo) -1 has a diagonal shape. Then, we compose C(+oo) with the reflection

S

=

thus yielding Cs(+oo) = S • C(+oo). Hence, there are trivializations satisfying = (-00) = c -00) (

and Ou(+ 00 ) = C(+oo), i,G,(+oo) = C s (+oo) or vice versa q5 u (+oo) = Cs(+oo), 1,b(+oo) = C(+oo), and which form an admissible pair by construction. So, to sum up, there is always an admissible pair of trivializations associated to (u, K) (v, L). The next lemma shows how to orient such equivalent operators simultaneously in a unique way by means of these trivializations.

(v, L) be equivalent Fredholm operators and let Lemma 3.8 Let (u, K) (çb0,1)) and (0' ,211) be respectively admissible pairs of trivializations. Moreover, let Det K and Det L be oriented by OK and °L. Then the pair ( b, 0) induces compatible orientations (b(oK)

7,b(o)

on the determinant bundle of the trivialized class e4,K4,-1 = ev,L,p-i if and that is only if the same is true for the pair (' , 0') on 00,14,-1 = e(oK)

'1,1/(0L)

•

Let us state separately the decisive step of the proof of this lemma as Auxiliary Proposition 3.9 Let e be a smooth --vector bundle and OK a fixed orientation for K E Ec . We denote the set of sections V) E C (End()), which are pointwise invertible, by GL. Further, let us choose any section •11) in Glic satisfying the additional asymptotical condition

(±oo) = id c (±,,) .

3.2. COHERENT ORIENTATION

117

Considering the determinant bundle Det eir on OK together with the orientation induced by OK, we are provided with an orientation 1,b(oK) for 1,b(K)

= 7,b K 0 -1 E eK

via the Hilbert space isomorphisms H 1 '2 (e)

H 1 '2 (e) and resp. 1,2 (C)

L 2 (4)

which arise from 7,1) and which are again denoted by J. Then this new orientation of the determinant bundle is compatible with the original one, that is

Proof. Resorting to the structure group of e, we reduce the given operator K E Ec and the transformation ip successively to a form which can be analysed

easily and explicitly. At first, we observe that we may start without loss of generality from the trivial bundle e = Ili X R'. Namely, if we choose any trivialization R x Rn , 7,b and K transform as Otriv = • • 10 -1 E GLt r jv and

Otriv (Ktriv) = 1,b(10triv •

Provided that the assertion is true for the trivial bundle, that is ( ° K))triv

71btriv ( 0 Ktr,v)

°Ktriv

5

we can transfer this orientation equivalence to the original bundle by means of q5, because it holds that oic oL

if and

only if Ç5(OK)

0(0L) •

The next step of the proof consists of showing that it is sufficient to verify the equivalence

any Fredholm operator L E T(H 1 , L2 ) which is homotopical with K, i.e. there is a continuous arc

for

' 2

: [0,1]

.F(111,2 L2

)

satisfying L o = K, L 1 = L .

Such an arc would then yield the equivalence 1,b(OK) COL) OL = OK due

to the vector bundle isomorphism

CHAPTER 3. ORIENTATION

118

Det(L, )

Det (7,b(L,))

[0, 1]

induced by 0. Hence, we may assume without loss of generality that the operator K E E tri„ is asymptotically constant and that it is admissible with respect to the gluing operation introduced in the last section. We shall now simplify the section E GLtriv = C(i,GL(n, R)) appropriately. Since we know from the assumption that //) is endowed with the end operators = 1 E GL+ (n,R) and that SO(n,R) is a deformation retract of GL ± (n,R), we are able to find a continuous path /P T from •//) to a 00 E C"OTI, SO(n,R)). Moreover, by means of a continuous asymptotical reparametrization, we may find a continuous arc ending in an asymptotically constant 0. Regarding such a 7,b E GLtriv , we can construct a similar gluing operation, so that, given

b , ;: b with 1/,(+oo) = ik—oo) and

K,

k

with K(-1-oo) =

we obtain the identity (3.7)

(0#7,b)(K#k) = 0(K)IkiRk) •

Since our orientation concept for Et riv is compatible with gluing, we now choose asymptotically constant operators L, M E Etriv satisfying AZ = 11, AT = A

,

4 = Ail and A-111-,f = .

Let us assume for the moment that we have already proven the assertion for the special case N E Etriv with 4 = 1 . (3.8) Then, due to the gluing operation, the general assertion is concluded from

o(L)#o(K)#o(M) o(L#K#M) (34) o((11#0#1)(L#K#M)) (3 7)

0(L)#0(i.1)(10) #0(14)

.

3.2. COHERENT ORIENTATION

119

Considering all these reduction steps, we may start without loss of generality from the special situation

/

cos

sin w(t)

1,0(0

— in cpet)

a at

0

cos ço(t) 1

K = — +1 and OW = 0

1

where cio E C"

S1 ) is asymptotically constant. We obtain the representation

IP(K) = 1P • K • 0 -1 =

a

(0(0 :-1 ) 1

a at

0

0

o-

1

= K - A.0 . Regarding the difference term Av, representing a compact perturbation of K, we calculate the estimate

(3.9)

1 6,011 C ( H 1, 2 ,L2

)

110110,2

<

CC

.

Given any r > 0, we define

so that the estimate (3.9) amounts to

1 — const . r

(3.10)

Therefore, provided that ro > 0 has been chosen large enough, the continuous arc

:

[0, 1] _> .F(H1,2 L2)

T

K

T Ao ro

runs merely through isomorphism operators H 1,2 know that

=

a +1: H',2 — at

L2

>' L2 , because we already

120

CHAPTER 3. ORIENTATION

describes an isomorphism. Finally, we see that on one side the transformation Oro maps the orientation [101' ] of K onto the orientation [101' ] of xo r-o, . On the other side, we can deduce from the homotopy .107:0 that both orientations correspond to each other by continuation. Hence, we obtain the relation

on,

(3.11)

OK

'-='• 011) 0 (K)

Oro

(°K)

Regarding the continuous homotopy with respect to the variation of r E [1, ro], we are led similarly te the correspondence of the orientations Viro (olc) and 7,b(oK) in the determinant bundle Det 0,(K) on [1, ro]. From this we conclude the assertion

CoK) = OK This result enables us to prove Lemma 3.8. Proof of Lemma 3.8. Let us start from the orientation equivalence

(3.12)

0(0K0 -1 ) = 0(0K) 0( 00 = 0(PLO -1 ) •

Then we have to verify the equivalence of the orientations

(3 13)0 /( 0K) = 0/ 0-1(0(0K0-1 ) )

and I/A(0 _ or i (0(oLik -i))

Moreover, on the one hand, we assume the asymptotical identities 0(-co) =tP(-oo) and 0'( -oc) = ipi (-00), so that

= 1 E GL(n,R) . Since on the other hand 0'0 -1 and 00,--1 describe curves within GL(n,R), the inequality det(001-1 010-1 )(+co) > 0 follows. Thus, we are led to the identity

(op,-

0/0-1 ) (+ 00) -

E GL(n,R) .

Altogether, we are able to apply the auxiliary proposition and hence derive the equivalence

(3.14)

0(0K) = (001-1 410-1 )(0(00) -

To sum up, the assumption (3.12) implies the assertion (3.13). By means of this last lemma, we now are able to transfer the orientation concept from the trivial bundle as treated in the first section to the Fredholm operators

{ (u,K) I K

E Eie`TM

in relation with the above equivalence relation:

3.2. COHERENT ORIENTATION

121

Definition 3.10 Any orientation of an L E K)1 induces a unique orientation of the equivalence class Ru, la, that is to say that given K L the equivalence OK

oL,

holds exactly if

0(0K) = Coi)

holds for any admissible pair of trivializations

Remark If we defined the equivalence classes to be of smaller a priori extent by the condition that trivializations with coincidence at both ends should exist, we obviously would be provided with more classes to be oriented coherently in the case of a non-orientable manifold. As we shall see from later features within the construction of a Morse homology, however, we do not have that much freedom regarding the choice of orientations for operators on curves which may be linked together yielding a 'Möbius band'. Actually, the dimension axiom will give rise to a condition on coupling between the orientations of two such classes. The crucial point worth mentioning is that we have already built in this condition by demanding the coincidence of trivializations at the negative end within Definition 3.7 of equivalence which might have seemed rather unmotivated to the reader. Assuming that we had fixed the coincidence at the positive end of the curve, we would be led to the 'wrong' dimension axiom, as shall be illustrated below in Section 4.1.4 on the example of 1F'2 (R).

Having accomplished the transfer to the non-trivial framework of varying curves, we now introduce a similar corresponding gluing operation by the trivialization method. We have to verify that this gluing is compatible with the trajectory gluing as regards the orientation induction by gluing Definition 3.11 First, we once again define the pre-gluing operation from the last chapter for arbitrary curves u,v E Ccx) (11, M) satisfying u(-Foo) = v( —oc). Given p po(u,v) we define

u(t u ro v(t)

={

p),

exp y

t-1

13— (t • e xp; 1 (u ( t + P + (t) exp 171 (v(t p)) )

)))

(+ 0

i ti

—

v(t — p),

t1

Here, the lower parameter bound po(u,y) is again derived from the asymptotical condition

u(t po), v(t — pp) E U(y) for t> — 1 and t <1 respectively in relation with the normal neighbourhood

U(y) of the exponential map.

Let us now consider trivializations 0: u*T

M : v*TM'=%1Eq. x Rn ,

122

CHAPTER 3. ORIENTATION

by starting first from a smooth trivialization

u(y) X W 1

F:

on the neighbourhood U(y). This induces immediately II-smooth tri vializations on the pull-back bundles u1(_ Loo rTM and v1[_,„1 )*TM, which may be extended to trivializations 0 and 1,b, such that the identities

E

(-1,..01, r va—

111[— co 0.)

are satisfied. Thus we are able to define a gluing operation for trivializations in a compatible way:

(kept') : (uepv)*Tm =

{

-9-=‘,111 x

0(t + p), t ..<, —1 ru#p,(t), It l 1 tp(t — p), t ?.. 1 .

By means of these trivializations we shall transfer the gluing operation for Etn, to the pull-back JR-bundles on the curves, but first, we have to introduce the

condition of asyrnptotical constance. Definition 3.12 Given K E

Eti.TM

and L E

Ev.TM, we obtain

K = OK 0 -1 and Lip = OLO -1 E

Etriv •

With regard to the time-shifting we notice the identity

(K p -= (K), , because the IR-action commutes with the partial derivative

a,

The next step is to define the halfway asymptotically constant operator associated to K Koes =

a

AK 0 + 0+ • A -IF(0

and L ip as analogously, having in mind the identity A LT0 the following relations are true: K oss and Lip L ip as in 9K g, and

A. Obviously, respectively.

Hence, the gluing operation already known on the trivial bundle for the operator K as #„L,p as induces the isomorphism

Det Ko„,,

Det

Det (Ko,„#Lip as)

3.2. COHERENT ORIENTATION

123

Thus. defining

Kdep L,p = Ko ,„#p L,p as for

p po =1 ,

we obtain an orientation of Kon,L ip induced by rp in a way which is independent of the construction of Kos and Loss based respectively on Ko and L,p, due to the homotopy invariance described in the last section. Moreover, since there is a T O such that

K4,#°p 1,,p(t) =

(Ko ) p (t), t s -T (L,p)_ p (t), t T

holds for all p po, the class 8R-0 # ; L,,, is also independent of the noncanonical construction of Kos and L,p as . Final/y, we may transform this gluing construction back to the non-trivial situation by means of the analogously introduced trivialization On*,

#(;)1, = (OK,10 -1 oepL.0)(44 °01, )

E E(u#p))*TM •

We notice again that, due to homotopy invariance, the class [urpv, K#°pL] and the associated orientation induced by # p° from o(ki, K1) and o([v, LI) are independent of all the above noncanonical construction elements. Summing up, we consider the noncanonical gluing operation

#0

[u, K , K+] x

[v ,

, L+1

[uep v,

L+] [u, K ±]#[v , L ±]

(u, K)#°p (v , L) = (uep v , K ep L) ,

which induces a unique and canonical mapping of orientations

(olu, K], o[v, LD

o[u, K]#o[t L]

with regard to the determinant bundles

Det ([u, K1)

[u, K], etc.

Due to the construction, the homotopy invariance and particularly the associativity may be deduced from the former trivial framework:

(3.15) ( o[u, If] # ofv, L] ) # o[w , M] = o[u, K] # ( o[v , # o[w, M]) . We shall round off this section by verifying that this orientation concept in relation to the gluing of arbitrary smooth curves u E C"(N,M) and Fredholm operators K E Eti oTA4 includes our former trajectories as a special case. Along these trajectories we consider the surjective linearizations in local coordinates

Du = DF„(0) : H 1 ' 2 T M) L 2 (u* T M), u E

Mx,y C CZY

'

124

CHAPTER 3. ORIENTATION

that is, the differentials of

=

exptï l o( 1 + VI) o expu : H I-2 (u*D)

L 2 (u*TM) .

at

In other words, the gluing #0 defined above is compatible with trajectory gluing as far as the orientation induction is concerned. Lemma

3.13 Regarding the gluing operation for time-independent trajectories X My,z 3 (u, v) 1-0 u#I,V E

endowed with fixed orientations isomorphism

Mx,z

o[u, Ra] and o[v,, Deb we observe that the

D#p : ker Dii x ker Di,

ker Dth#,„

induces the same orientation o[u# Du# ] as we obtain from

olu, Dul#olv, D„j . Additionally, the identity of the classes

[u# ,

= tuigt;v, Du#°p Dd

holds. Proof. Let us assume any fixed pair of trajectories together with compact neighbourhoods N(u) C Mr,y and N(v) C My,z. Given the compact set K = N(u) x N(v) we choose a fixed gluing parameter p ?,- p(K). Then the glued trajectory is represented as

u#pv = expu#7,v (n, y, P) . Let us now consider H: K x [0,11

(u,v ,

T)

M) C Px1 ;2z exPu#p,

'YK(U,V1P))

yielding the homotopy hr = H ( T) from ep to #p. Provided that the gluing parameter p has been chosen large enough, the differential

v) : ker Du x ker Dv

H 1 '2 (hr (u, v)*TM)

proves injective for all T E 10, 11. This follows from the deductions in the last chapter in the same way as we obtain the surjectivity of Dh,(v ,,,) and the isomorphism relation

(3.16)

Proj L2 ker ph, o D : ker Du x ker Dv

ker D

3.2. COHERENT ORIENTATION for all r

E O,

125

1]. As a consequence, the isomorphism

D#p = Dhi -=Pro.V: Ke2r Dhi o Dhi induces the same orientation on

[u#pv, D u#,v] = [ule p v, Du#p7] as PrO iter 2

Dho Dho = Proj

1310; °

Din •

According to the construction of the pre-gluing map in the last chapter we can find admissible trivializations of the pull-back bundles u*TM and v*TM, such that we are led to the linear mapping II' : ker Du ,triv X ker Dv,triv

ker Du#;v,triv PP:4:r D u$fr; v Ariv

( 61

on x R. We denote the operators from

Etriv

K = 13,,triv and L

+ 0+(-p)

by

Dv,triv

throughout the remainder of the proof. It remains to show that we can homotope this trivialized version 4) continuously to the gluing version used in Definition 3.12 above. Actually, we may construct suitable homotopies from K and L to respective halfway asymptotically constant operators Kas and L as, which exclusively go through surjective operators. 5 Hence, we find a homotopy from the map 4) to the isomorphism ker Kas #p L as

ker Kas x ker L aa ()

Hi

Prot:,

+(_p)

which similarly goes merely through isomorphisms. This is in fact possible, because the compact interval of time around 0, on which the curves AiK.,4 p L„. and AD„#0,„triv from Cc° (R, End(Rn)) differ, decreases as the gluing parameter increases. The central element of this homotopy through isomorphisms is given by a continuation of the form T

— T)

±

TI 6p

[( 1

T)

0+ ± T] (- p

Actually, the proof that the corresponding linear gluing version is an isomorphism may be accomplished according to the scheme of Proposition 2.50 for each parameter T. Hence we conclude the proof of the lemma. 0 5Note that the surjective operators form an open subset such that suitable asymptotically constant operators can be found.

CHAPTER 3. ORIENTATION

126

Remark Similarly, we may state a corresponding result for the gluing operation for mixed broken trajectories, now based on a linear gluing version of the type

(,p)

01 p • ± 0 + p • 2 p

and i3

2+i3p+ • Ç

respectively .

We can reduce this result to Lemma 3.13 by means of a continuous variation of the form )9

01-p . ( 1+1- ) p

)

04 . ((- 1+1-)p, rE

11 •

Orientation and Gluing for A-Parametrized Trajectories Let G')3 : [0, 1] x P

JJ -4 L 2 (P!•2,y0 *TM) 1

G'f3 (A , •-y) =+

be the mapping

V I-1') 3 0 -y

N/

associated to the A-homotopy 11"'3 : [0,1] x M -> R. Given the local coordinates (exp„, V2 expu ) at (A, E ./k4 ,y0 = G')3 1 (0), we consider the corresponding linearization DG') 3 (A, ILA) : T),[0 , 1] x H 1 ' 2 (u*TM) -> L 2 (u*T M)

(r,)

DI G° (A , u),) • r + D2Ga (A,u,x)- •

As we know from the transversality results, DG')3 (A, /LA) proves surjective for a generic metric. The Fredholm operator Du, = D2G')3 (A , UA) E E u •T m with the index 1.1(x a ) - p(yo), however, does not need to be surjective at all. This can merely be provided as an a priori statement for the boundary parameters A = 0,1. Hence we conclude that DGO(A, nA) is a Fredholm operator with index /L(X) - tz(y o ) + 1 for all (A, uA) E

We now wish to compare the gluing embedding ra,YO

x

Mt° 110,z0'

x

k MH

y.,zo

resp.,

for A-parametrized trajectories with the more general gluing of Fredholm classes with regard to the orientation induction. At this point, we once again have to resort to the fundamental sequence Lemma 3.2. Dealing with the orientation problem is crucially simplified by the determinant bundle and the gluing with respect to classes of Fredholm operators [K] always being led back to the respective surjectively extended operator kip Rk x _, L2 as far as the concrete construction is concerned. But this exactly concerns the relation L2. Let between the operators DGa/3 R x Hi.2 _, L2 and D2G "i3 Hi.2 us consider the following short exact sequence: :

(3.17) 0 —4 ker D2G"'8

d2 T), [0, 1] ker DG"fi —4

coker D2 G''8 —4 0

COHERENT ORIENTATION

3.2.

127

where d 1 ( ) = (0, ) d2(T, = d3(T) = [DIG° r]R(D2 c.p) •

According to the fundamental Lemma 3 2, it induces a canonical isomorphism A"ker D2Ga [3 (A, UA)TA[0, 1]

(3.18)

A"ker DG (.\, u),) for all (.\, UA) E

Amaxcoker D2Ga [3 (A, u),)

This amounts to a canonical bundle isomorphism Det D2 G')3 T* [0, 1]

Am"ker DGa)3

on the components of the parameter-trajectory spaces M". Thus, the choice of the fixed orientation _= 1 on T. [0, 1] ll gives rise to natural isomorphisms of the above line bundles on these manifolds of )-parametrized trajectories (3.19)

: Det D2G")3 =4 ‘ A'ker DGa)3 .

Given any pair (A, u),) E .A4 x11 , yo , 5/ uniformly induces an identification of an orientation of the operator DI, E E U TM with an orientation of the tangent space T(A,,,,).A4

yo .

Proposition 3.14 Let u.s consider the gluing operation for )-parametrized trajectories

.A/1

,yo X .A4

/3

,zo

3

((A, 70,), v) 1—) (A, u),)# pli v = (3%, w5;)

together with given orientations otio„ Dr,,,] and o[v Dd. phism ,

E Ni x ilc„zo

Then, the isomor-

„ f13 AA II X Tv./v, zo — 4 TA,u1)11vd"` x.,zo

D# pIl :

in connection with the isomorphisms fl(A,u ),)

SI ((A, to,) # p// v)

Det-D u , —4 AmaxT(A,u1) m x.11. ,y Det Dws,

,

A max T(A,u1)4q,1vMx11,,,z o

induces the same orientation 0[11), Dr,] as we obtain from the gluing o[u)„ Dr,] # o[v, Dr]. Here we use the identity of the classes of glued operators [w , D i ] = [uAiepv, Duleppv] •

CHAPTER 3. ORIENTATION

128

Proof By analogy with the proof of Lemma 3.13 we regard homotopies and trivializations at suitably large gluing parameters, so that we can reduce the assertion to the following special case. Let K, L E Etriv be halfway asymptotically constant with matching ends K+ = . We further assume that L is already surjective and that R —> L2 is a linear mapping, such that

R x 11 1 '2

L2

(T , e)

71) • T

K•e

is surjective, too. Then, defining koitp/I L : R x H1 •2 —* L2 by

(k,,/,# p1/ L)(a, e) = with

ti) • a + K # ,t,iL • t p AK(t), AL(t — 2p), t p

AKw(t)

and choosing p large enough, we deduce that

ker (ktp#

t9p : ker kip x ker L (3.20)

(a, e,

L)

1—* Proj r—" L2 (a, e + (_2,) L

is an isomorphism. The proof is accomplished by exactly the same steps as are presented in [F-H] for the more general version of the gluing of asymptotically constant Fredholm operators. By means of the homotopy of the form

(e,

e(1-1- )p -4- C(-1---r)p) T C [0 1 1 ]

7

we are able to transform the actual linear version of gluing into the shape used in Proposition 3.6 for any lower parameter bound pl that is large enough. Therefore, without loss of generality, we may start from the version

(3.21)

: kerk,j, x ker L ((a, u), v)

ker (ko# p L) 1—*

Pp(a, up + v_p)

and compare it to the linear gluing version

(3.22)

(bp : kerk, x kerZ, -ker (k,#)

used in Proposition 3.6. Considering the isomorphism

Am"ker kip

Am"ker L

Am"ker k/f pL

induced by 'Op , we observe that due to the identities ker k, = ker Ky, for

Ky,

Rx

R x L2 (a, u) 1—* (0, k v,• (a,u))

129

3.2, COHERENT ORIENTATION and

ker ko#pL = ker (K#A),/,,90 we may also state it as Am"ker Kt',

A'ker L

Am"ker (K# p L), c, .

Thus on the one hand, the choice of the canonical basis the isomorphism

(3.23)

Det p : Det

Det L

1* E Rs gives rise to

Det (K# pL) opeo

induced by t 9 p by analogy with the deductions in Section hand, (1) p yields the isomorphism

(3.24)

Det p : Det K

Det L

Det

Using the natural isomorphisms Det K Det K we are led to the result that Det /9p as well as Det morphism

Det K ® Det L

3.1.2. On the other .

,

Det L Det L,i„ etc., p induces the same iso-

Det (K# p L). Since the composition

A'ker k,, = A'ker K 1, ,,

A'ker K , 0 R* = Det --+ Det K ,,1

is expressed exactly by the natural isomorphisms defined above, Sr i

Am"ker

Det K ,

the proof is accomplished. Coherent Orientation Up to now we have developed a concept which enables us to join Fredholm operators K E Ext*TM along compact curves u E C"(R, M) into equivalence classes, which may be oriented as one with consideration of the gluing operation. The last step toward the key feature of our intended homology theory consists of uniformly orienting all those operator classes in a way which is compatible with gluing, i.e. in a coherent way. Definition 3.15 Let A

= { [u, K] I u E C cc (rt, M), K C

Eu•TM

be the set of the equivalence classes of Fredholm operators along smooth compact curves. Then any map a, which associates an orientation cr[u, K] of the corresponding determinant bundle Det ([u, If]) [u, K] to each class [u, K] E A, is called a coherent orientation, if and only if it is compatible with the gluing operation

cr[u, K]ita[v, L] = cr[u#v, K #L]

CHAPTER 3. ORIENTATION

130

for all pairs [IL, K] and [v, L], which are admissible for gluing, that is, which are endowed with matching ends u(+co) =- v(-co) and K+= L- . Let us denote the set of all such coherent orientations by CA. Moreover, we are interested in the group

r=ffE0-1,-11"

I nu, K] # [y, L]) = nu, Kl)

, L])

with respect to pointwise multiplication. By the following proposition we verify that this definition makes sense. Proposition 3.16 Coherent orientations exist, i.e. CA 0 0, and the group acts freely and transitively on CA by

r

rx CA CA (f, o)

f

(f, a) ([u, Kl) -= f

Kl) • aju,

.

Proof In the first part of the proof, we present an algorithm for constructing such an orientation. It is crucially related to the associativity verified above (see also [F-11]). Let x0 E M be an arbitrary point in M and let A G End(Tzo M) be non-degenerate and conjugated self-adjoint, for instance X0 G Crit f and A = H2 f(x0) for a Morse function f. Then, we consider uo E C(R, M) to be the constant curve up

x 0 = const

together with the operator

K0 = + A E r — 14TM which is an isomorphism as A E C'(End(4TM)) is constant. 6 Thus, we are R*. able to fix the orientation 1 0 1* for the determinant space Det K0 = Briefly, the class [u0, K0] E A, called the anchoring class, is oriented by [1011. The next step is to consider the subsets

S - = { [u, K] E A I u(-oo) = x01 and

5+ = { [u, K] E A I u(-1-oo) = X }

of A and to choose an arbitrary orientation c[u, K] for each class [u, K] [uo ,K0] = [u0] from S - . Considering the one-to-one correspondence S - S+ induced by the reversal of the time-parametrization, that is,

[v', L'] E S — for 6u0 = const also implies the identity Vt =

[v, L] E

3.2. COHERENT ORIENTATION

131

with v 1 (t) = v(-t), A L -i(t) = AL(i) ,

we consequently obtain unique orientations on the classes [v, L] E S+ already fixed by the condition = a[uo] .

Obviously, this proves consistent in relation to the anchoring class [uo ] E S—

n

a[uo]#a[uo] = a[uo] . By means of these two types of classes from S - and 54- anchored at xo together with the associativity of gluing as far as the orientations are concerned, we are now able to accomplish the last step toward a coherent orientation. Considering an arbitrary [u, K] E A, we find unique classes [va , Ko , K - ] E S — with

vu (-1-00) = 24-00

and

[wu, K + , Ko] E S + with

wu (-oo) =

Then, an orientation a[u, K] is determined uniquely by the condition

a[vu , Ko K - ]#cr[u, K]#Gr[wu, K+, Ko] = a[uo] which is well-defined due to associativity. Moreover, associativity guarantees that this construction in fact provides a coherent orientation. We now analyse the ['-action; Given two arbitrary coherent orientations ai and a2, we define

f: A

{±1}

by

f • 0-2 •

Obviously, this is equivalent to a2 =

f•

al

Hence, given two classes [u, K] und [v, L] admitted for gluing, we are led to the identity

al[u, K]#ai[v, L] = ai [u#v, K#L] = f [. • • # • - • a2[• • # • .] = (f cr2)[u#v, K (1. . .# . . .1) • (c12[11, K]#(72[1.), L])

= f ([• • • # • •])(f[u,K]cri[u,K]#f[v , Lia [v , L ] )

132

CHAPTER 3. ORIENTATION

This amounts to

1 = f [u#v, K #I1 - f [u, K] f [v, LI . Applied in both conclusion directions, this means that such a mapping f : A --4 {±1} represents an element of the group F, exactly if it transfers coherent orientations again to coherent orientations. Hence we conclude that the asserted group action is in fact well-defined and transitive. Finally, it is straightforward to verify that this action is free. 0

Chapter 4

Morse Homology Theory 4.1

The Main Theorems of Morse Homology

We introduce Morse homology theory by two main theorems about the existence of a canonical boundary operator associated to a given Morse function f and about the existence of canonical isomorphisms between each pair of such Morse complexes. These theorems appear as the essence from the theory on compactness, gluing and orientation which has been developed throughout the last three chapters. It is the isolated trajectories of the time-independent, time-dependent and the )-parametrized gradient flow which form the crucial features in the core of these main theorems. The following preparatory section describes how we associate respective characteristic signs to these isolated trajectories.

4.1.1

Canonical Orientations

Concerning the Fredholm classes [u, If ] E A analysed in the last chapter, we henceforth denote for all trajectories u of the time-independent, time-dependent or .A-parametrized gradient flow by [u] the class [u, Du ], where Du, stands for the surjective' Fredholm operator yielded by the linearization of

+ Vf,

etc.,

in local coordinates at u. Due to the surjectivity of these operators together with the representation Tit .A4,sy kerD, c H 1 ' 2 (u*TM) lexcept for the A-parametrized gradient flow; see below

134

CHAPTER 4.

MORSE HOMOLOGY THEORY

for tangent spaces of the trajectory manifolds, an orientation of the class [u] gives unique rise to a differential-topological orientation of the connected component of M s ,y containing u. Now, in contrast with the notion of a coherent orientation, we find the following canonical orientations for the classes [u] E A in the cases of isolated trajectories u: • As to the time-dependent gradient flow, i.e. for isolated trajectories in ker Dy, vanishes, so that we may orient the trivial line bundle M z—so hœs Det [u] = JR 0 canonically by 1 0 1*. • In the case of the time-independent gradient flow, i.e. u E ker Du turns out to be of dimension 1. We find a canonical orientation by Oit = — Vfo uEker Du ,

because these solutions of an autonomous differential equation are endowed with a one-dimensional shifting-invariance. This orientation o([u]) is briefly denoted by [ut]. • As to the A-parametrized gradient flow, we consider the natural bundle isomorphism from the last chapter

: Det D2 G°

(4.1)

Am"ker DWP ,

which was introduced on components of the parameter-trajectory spaces

M x11,100 , where D2G"3 (A,u,x) = DUA E EtTm, u E M x11„ y is generally not a surjective operator. Restricting the analysis to a relative Morse index p(x) — u(yo) -= —1, this means dim ker Du , = 0 and dim coker Du , = 1 and therefore isolated A-trajectories, we obtain by f2 the orientation 1 0 (Di co. A)* on Det D2G° = Det .D„), from the canonical orientation 1 on A"ker DG QI 3 = R. Here, we apply the canonical identification

coker D2G''° (A, u),) R(DiG°(A, ux)) . We deal equivalently with the canonical orientations I> 1 0 1* on T(À,,,, À )M x/1. ]yo and

1 0 (DiG'S(A,u),)• A) * on

Du,,1

If we consider the 1-dimensional components of Msif , y0 , i.e. p(x) = p(yo), and regular A-values, which means regular with respect to the projection map

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

135

so that D2GaP(A, u),) = Du, is onto, the natural isomorphism

Det

T(„u,)M

vo

identifies the canonical orientation 101* on [D,,,,) with that orientation on A47 which is mapped onto the canonical orientation A, of [0,1] by the projection map r. Deciding whether two orientations [1 0 and [1 0 11.„, 2 of operator classes [D.„,, ] at regular A-values and within the same component of MY. yo are equivalent amounts to analysing the pre-image of the projection Map on the interval of A-parameters; see also Figure 4.4 on page 146.

Definition 4.1 Given any fixed coherent orientation a as developed in the last chapter, we are able to associate characteristic signs rc,(u) to the isolated trajectories by comparing the above canonical orientations for [u, Du ] with a, with respect to the three different types of isolated trajectories, that is 1. for u C M

:

Ta (IL)

-= [tit]

2. for u E M xh: °,,0 : Tc (u) a[u] = [1 0 V]

Ha"3

3. for u E M x À,vo : r(u) c[u] = [ 1 g (D i Cao(A, u) •

4.1.2

.

The Morse Complex

Definition 4.2 Let f E

(R, M) be a fixed Morse function and

Critkf = {x E Crit f p(x) = c}

for

0 -<,k n

the discrete subset of M containing the critical points off with Morse index k. We define

C k f = Critkf 0 Z as the free abelian group generated by the critical points with Morse index k, i.e. in particular Ck(f) = 0 for k < 0 or k > n. Given a fixed coherent orientation a E CA on M we now define the mapping

• ) : Crit f x Crit f

Z

by

(4.2)

(x, y)

E

T(u),

for

p(x) — p(y) = 1 ,

A70- 11,„

o,

otherwise .

CHAPTER 4. MORSE HOMOLOGY THEORY

136

The pairing (x, y)u amounts to counting the connecting orbits with relative Morse index 1 between x and y equipped with their characteristic signs from the above definition. Here it is worth mentioning that the characteristic number (x, y)U corresponds exactly to the geometrical intersection number of

Wu(x) rh Ws(y)/R as described in the introduction, up to a suitable change of the noncanonical orientation of the respéctive critical points. This noncanonical ingredient is equivalent to the noncanonical choice of a coherent orientation. We shall illustrate this interrelation between the present approach to Morse homology and the classical approach by Thom, Smale, Milnor and Witten in more details in Appendix B. Since Critkf is the generating set of the group Ck (f) for k = 0,...,n, we extend ( • , ) to

( •, • )Gr : C(f)®C(f) —> Z

(4.3)

and we define a homomorphism sequence (aZ)k=o ,...,n associated to (Ck)ko, = ...,n by : ck(f)

!3x=

ck-i(f)

(x, y)'y

yECritk- f

This definition leads us to the first main theorem within this work on Morse homology: Theorem 7 Given any Morse function f and any coherent orientation a, the family of homomorphisms (ck(f) , aZ) kEz = (c.(f) , represents a chain complex, i.e. it holds that

8_ 1 0

0 for all k =- 0, , n

Within this monograph, we call this chain complex the Morse complex , in contrast to classical theory, which denotes by Morse complex the associated cellular complex2 . Proof Given any x E Ck(f), i.e. without loss of generality x E Critk f and 2associated to

f

by homotopical equivalence with

M,

see [M11

4.1.

THE MAIN THEOREMS OF MORSE HOMOLOGY

137

k 1, straightforward computation according to the above definition leads to the equation

E

a2 x

E

(x, Oa( y, z)°z

zECritk-il yEeritk-if

E

zECritk-il

Ta(fi) •

Tam

z,

where we denote by M x,, the set of simply broken trajectories between z and z, as was defined in the gluing Section 2.5. Now, resorting to the cobordism equivalence, we may reorder the double sum on the right hand side, obtaining

(4.4)

02x=

E

E

zECritk_2f

Ta (Ü)

• Ta (i ) z .

(ii,i3)E1(11,)1

Since we already know that each of these equivalence classes contains exactly two different broken trajectories, we could deduce the assertion in the special case of Z2 coefficients from the identity

E 1

E-

0 mod 2 .

The analogous general step concluding the proof,

E

(4.5)

7-„(i-,) •

Te

(i3 )

=0

,

will be proven in the following lemma by means of the interrelations between gluing, coherent and canonical orientation, i.e. briefly by oriented cobordism theory for simply broken trajectories. Thus, the theorem is concluded from

(4.5).

Lemma 4.3 Let

s, y, y', z E Crit f be critical points satisfying p(x) — 1 = P(Y) = p(V) = A(z) ± 1,

and let i1) E

x flithz , (112, 4)2) E M

c,

,

X

y',z

be equivalent broken trajectories, i.e.

f/i) es, (t12, 02)

within

Then, with respect to any arbitrary coherent orientation a on M, the identity ra(ii 1) To0i) = --Tcr(fl2) • 1-.(i2)

holds.

CHAPTER 4. MORSE HOMOLOGY THEORY

138

Proof. The cobordism-equivalence signifies per definition that both broken trajectories are mapped by the respective gluing operation # into the same component of the unparametrized trajectory space M x , z , which is diffeomorphic to (-1, 1). The strategy of proving this lemma is to compare the orientations which are induced by the gluing operation on this component from the respective canonical orientations. The following relation between the canonical orientations defined above has to be verified: (4.6)

t]

t] = — [112 t] # [2 t]•

Then, according to the definition of rc, (ill), etc., the assertion is concluded from the coherence condition and the cobordism-equivalence: CrECA

0171 1] # P[I] (

(

21 ,f)1 ): 22,.0 2) CrECA

Hence, we have to analyse how the canonical orientations behave with respect to simply broken trajectories in relation to the gluing of trajectories. We consider the diffeomorphism from the time-shifting action

.k7fx ,y x R (û, r)

(4.7)

in relation to both versions of the trajectory gluing, for parametrized as well as for associated unparametrized trajectories. Let f)) be a fixed unparametrized broken trajectory from the connected component Mr,z ° , Po an appropriate gluing parameter and E> 0 fixed, so that we obtain the following commutative diagram, ffil(_,. ) X

10(_)

#po

IR (Po

— E Po ± E)

fvf—s,z °

together with the embeddings

and

P

(fi(9—P 0 ) , f) (P0 -9))

191— *

Op?)

Note that the appropriate choice of po depends on E. Now, let : .Aqx , z ° f", (-1, 1)

THE MAIN THEOREMS OF MORSE HOMOLOGY

4.1.

139

denote the diffeomorphism which endows the one-dimensional connected component with a fixed orientation vector, e c To( poiv—I x , z °, i.e.

a DO • e -= — as

E Tsa (-1, 1) .

As a consequence, time-shifting (4.7) and the orientation [e] on .A4 = A4x, z 0 /R induce a fixed orientation (k, e) on A4., z °. On the other hand, we may also consider the canonical orientations on X {f)} ( _

)

by

ei-L —0 ) ,

where the embedding (,.9 identifies the pair of tangent vectors

,

E

exactly with

—2))

E T010 ,

(Po — 6 1 Po +

(..A4x, y X

f).

Referring to these fixed orientations, we now have to determine whether the embedding #po acts in a preserving or in a reversing manner. According to the above commutative diagram, it identifies the pair of tangent vectors Ca, —0 with the pre-image of the vector DV; • g with regard to the differential of the projection map IR. It likewise identifies the pair ea, 0 with the vector 0 E Tfi# po b-M x , z ° canonically induced by time-shifting. Hence, the question of orientation preserving or reversing by the gluing operation #po with the tangent vector on is reduced to the comparison of the fixed vector (-1,1) which is induced by o Vi together with g c Tpo (po, -E, po+ f). Exactly at this stage we derive different characteristic signs for the two equivalent broken trajectories (al, fh) and (02, 02).

+.

a

Assuming the convergence liM 0(i14 pf)i) = 1 P-403 and respectively

lim 0 (û41)2) =

—1

p-00

for a given fixed diffeomorphism 0, we observe that •

Q by .01(.) = 01#.01

• -a

and

by '02 (') = 1L2#112 and

and with .

g

is identified with

140

CHAPTER 4. MORSE HOMOLOGY THEORY

•y t

Figure 4.1: Cobordism-equivalence for 0-trajectories Thus, we deduce the relation asserted in (4.6) between the orientations induced by gluing of equivalent broken trajectories. This adverse orientation induction is illustrated in Figure 4.1. The arrows on the broken trajectories mark the canonical orientations (ki t ], —kid), which under the gluing operation correspond to the respective orientation of the one-dimensional connected component .A4,,z° according to the respectively indicated weak convergence of unparametrized trajectories. 0

4.1.3

The Canonical Isomorphism

Given two Morse functions, we now intend to derive a chain homomorphism between the associated chain complexes Cs (f") and C. (f 13 ), which gives rise to a canonical isomorphism between the respective homology groups. Actually, we will obtain this chain homomorphism in the same way as we extracted the canonical boundary operator from the negative gradient flow of a fixed Morse function. As to the isomorphism, 'canonical' means the independence from any further assumptions apart from fa and fl) . Definition 4.4 Let fa and ff3 be any Morse functions on the manifold M, let had3 be an arbitrarily associated Morse homotopy and let us choose any coherent orientation c r. By strict analogy with Definition 4.2, we define (

Trcro

( Ck(f a ))k=0,

,„ x

(C k (P))k=0,...,n

Z

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

by

E

(x,,„ xo )°,03

(4.8)

141

r(u), for il(sa) =

0,

otherwise

x Crit f 0 . Once again, the compactfor the generators (x,„, xo) E Crit consisting of isolated trajectories trajectory space M h x : ness result for the guarantees the finiteness of this sum. Consequently, we define the group homomorphism.s by analogy with the above ak, that is 4'k(ha13)= tdZ a :Ck(f a) =

Ck(f'3)

E (xc„ xx, .

xpECritkf°

Now, the fact that the latter sum is finite is derived from the compactness results for Morse homotopies as was stated in Corollary 2.46. Note that this relies essentially on the coerciveness of the considered Morse functions. The first step toward the second main theorem of Morse homology consists of the verification of the fundamental functorial relation for chain complex

rnorphisms: Proposition 4.5 The family of morphisms 41); a

= (4311

: a

Ck

(f a )

Ck(P))

forms a chain homomorphism :

C4,(fa

C(f) ,

i.e. it holds

arc (f 0 )o (1)ft` = (1 Proof. The

1 0 az(fa) for all k = 0, ...,n .

proof follows that of Theorem 7. Using the short notations 93

aa =ar,(fa),

we compute OS 0 4)13a

—

0

a.)(xa)

E

E

fa(x0)=k

14.0EM. ct ,. 0

ao(

E n(xa, yo) yo p(10)=k-1

(Uco) Xi3)

aZ(f)3 )

142

CHAPTER 4. MORSE HOMOLOGY THEORY

where

n(x,, yo) =

E

E

P(x)=k uEM

E 5

7-a(nas)•

To 0710 )

itpaig,"

—E E E )= k - 1 i•Lor

,vo v,p EM : 13 ,,,0

We now prove the identity

(4.9)

n(x,„ y0) = O.

in a way that is comparable with Lemma 4.3. The analysis of the cobordismequivalence for mixed broken trajectories with fixed endpoints

(xa , ya) E Crit k r X Critk-1 f f) this time requires a separate discussion of two different cases related to the order of the trajectories:

0 x (a) (uao fio ) E .A,111° sc„,x0 ,

(b) (fia , vo) G

x 13,y0

and

x Mh' d3 4/13 •

Starting from the broken trajectory (uao, ils), we observe two different possibilities for cobordism-equivalence with regard to this distinction of cases:

(a)

x Mx, (uao, fio) for a x's G Critkf °-. (wao, '17)0) E Analysing the orientation induction by the gluing operation for mixed broken trajectories, we are led to an identification of the canonical orientation ( [1 0 V], [fio, t ] ) on the component of (van, ito) in x .Pxo No with that orientation on the one-dimensional component M rc„ yo °, which corresponds to a decreasing parameter p regarding the gluing of broken trajectories. Actually, we observe this orientation correspondence at both ends, because we deal both times with broken trajectories of type (a) according to the above specification (see Figure 4.2). Hence, this case leads to the identity

(4.10) (b)

Ter (V a0)

(fiii) =

(

WO)

'

eel0) •

(nap, fio) (ûa , vas) E x Mv.,yo for a ya E Critk_ i f a . In the case of an equivalence with a mixed broken trajectory of type (b), this trajectory is endowed with the canonical orientation

01 0,,t1, ri

1 1) ,

143

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

;,3

Figure 4.2: Cobordism-equivalence for mixed broken trajectories of type (a)-(a) which now corresponds to an increasing gluing parameter (see Figure 4.3). Thus, the gluing operation induces at the cobordant broken trajectories of different type the same orientation on the associated one-dimensional component .A/I s‘.,1,0 °. Hence, we verify the relation

(4.11)

(Ua L3) • To.

Via)

= To.

efi ci ) •

Ta (Va

s)

in this case. As a consequence, we obtain products of characteristic signs with different signs in the case of mixed broken trajectories of the same type as in (4.10) and with equal signs in the case of different types as in (4.11). Since the cobordismequivalence from the compactness-gluing-complementarity yields a one-to-one correspondence between mixed broken trajectories regardless of type, the above case distinction accomplishes the proof of the asserted identity n(x„, yo) =- 0 and thus of the proposition, too. The next step now consists of verifying that the induced homomor-

phisms on the level of the homology groups, 4>e- :11.(r)-,H.(fo) are in fact independent of the choice of the actual Morse homotopy f

h..43

f0.

144

CHAPTER 4. MORSE HOMOLOGY THEORY

yI Figure 4.3: Cobordism-equivalence for mixed broken trajectories of type (a)-(b) Proposition 4.6 Let hr and ha i S be two regular Morse hornotopies between fa and ffl together with their associated chain rnorphisms and 4)113. a . Then there is a family3 of morphisms

(vga

111 1 1 '

Ck(fP), k E Z

satisfying the identities (4.12)

(It — 447k = 4+1. 14ca

4»Za 1 o aicct

•

In other words, represents a chain hornotopy between the chain complexes C(fa) and C.(f3 ).

Proof. At this stage we finally have to apply the results about transversafity, compactness and orientation as far as the spaces of A-parametrized trajectories are concerned. Let us choose a regular homotopy

h0

11' 43 — 1

with the homotopy parameter A, so that, due to the regularity result in Theorem 2, we obtain the finite-dimensional manifold

A4 x.,Yo B-43 = 3We extend

1 '2 (A, u) E [0,1 ] X PXa

the finite family

C.(r) by C,(f

(A) }

I

UE MH " ,Yl3

=0

for j
> n.

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

145

associated to the fixed endpoints (x cc , y0) E Crit fa x Crit f13 . Here, we use the brief notation M x„,y o (À) for the trajectory space associated to the fixed parameter A. 4 The ascending of the degree of the operator Cs

(fa)

C5+1(f'9 )

is due to the fact that, within this framework, we obtain isolated trajectories exactly if the relative Morse index is

Under this condition the finiteness of the set .4411:3y0 is provided by the respective compactness result. Again, these isolated trajectories can be counted with characteristic signs and we define k Ck (fa)

Ck+1(P)

for k E Z on the generators x c, E Crit fa by

E(xa,z,oz

Wka)3xc, (ra

,

zs ) =

E 'r(u) .

Thus, we have to compute the difference of the terms

E

490 o xli:a (x a) =

E

Ta (U),) 49)3 Z0

A(z0)=k-1-1 (A 71, x )Em11:10

(4.13)

E T,(u)., 1,3)]

=

(

P( S 0) =IC 0( Z 0) = k+ 1 Pt ,U X)EMC I: 0_ 11,3E1C/i f ° .0,r0 0,

and

v7,!1.49-(x.)

=E

E

Tcrol.)410-y,,,

1-411.)=k

(4.14)

E

[

E

E

(s)=k 1.1(y.)=k-1

E , -ia ) • Ta (VA)] (

X0

(À,v),)EMV,!),

with regard to the right hand side of equation (4.12) and (4.15) (DP — (13g') (x a ) =

E

ET

A(SO)=1 4 (X.)uiEM%, x0

4This is not necessarily a manifold!

(U) —

E To(uo)]

U0EM22„.0

X0

146

CHAPTER 4. MORSE HOMOLOGY THEORY

with respect to the left hand side. In principle, all steps which have to be gone through are known from the above propositions as applications of cobordismequivalence for trajectory spaces. In this framework, however, we have to apply our results on compactness, gluing and orientation to the 0- and 1-dimensional manifolds of the A-parametrized trajectories. This leads to a slightly richer variety of possible cases to consider. This time, the connected components in question may also be manifolds with boundary. Hence, regarding the cobordismequivalence, we have to take into consideration more separated cases. The sketch in Figure 4.4 suggests a geometrical visualization of the closed submanifold C [0, 1] x P1 ,2,zo for i(x) = p(zo). There, the ends of the one-dimensional component.; which have a boundary in the weak sense of convergence toward mixed broken trajectories, are indicated by open brackets `r. p1,2 x,y

A

1 Figure 4.4: The 1-dimensional manifold with boundary of A-parametrized trajectories With regard to the components of the one-dimensional A-parametrized trajectory spaces, we have to consider separately the following diffeomorphism types:

(a) [0,1], with OMY (b)

SI,

(c)

( — Do,

(d)

[0 ) 0.0)

CO),

and (—oc, 1].

C Mx—xo (0) Mr,„x o ( 1 ),

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

147

Obviously, case (b) does not play a role in our cobordism-equivalence, because those closed components neither contain boundary points in the strong sense, that is, curves u E M, 0 (0) Li Mx c, ,x0 (1) as trajectories for the timedependent gradient flows yieldirig .A4 x/1°..x0 arid .A4 sh o,1 nor boundary points in the weak sense, that is, mixed broken trajectories, composed of trajectories vA E M x ,,z ,(A) satisfying tr(x') — 1.1(.z i ) = 1 and respective a-trajectories. Concerning the oriented cobordism-equivalence for trajectory spaces, the gluing operation associates to each mixed broken trajectory —

(u)„ fio)

e .A4 z_ zo (A) x .A4f:30 , 0 and (ica , v),)

e MÇ

x

building up the terms in (4.13) and (4.14), a one-dimensional connected component of .A4g of type (c) or (d). Additionally, regarding the terms in (4.15), the trajectories uar3 E M s„, x0 (0) 1-.1 Mxso (1) lead us to connected components with boundary in the strong sense, that is, of type (a) or (d). Hence, the discussion of the cobordism relations in the proof of Proposition 4.5 has to be extended by the cases (a) and (d) of manifolds with boundary.

As to (a): The connected component A/G-Ic. so ° in question is diffeomorphic to the compact interval. As we analysed in the first section of this chapter, the natural isomorphism S1 induces the following orientations of the component from the canonical orientations [1 0 1. ] of the classes [ud, u, E M xh *y , i = 0, 1, at the boundaries of the interval, A = 0,1: • A = 0 -.4 orientation toward the interior of the component, • A == 1 -.4 outward directed orientation. Given the boundary curves

a ./v1 Hzar ,zo ° = {

} C M xh °,, so U Mi sa ,z0 h

we consequently have to distinguish the following two cases:

1. Assume fu oco, c .A4xh 0 13 or .A4%,x0 . Then, in both cases, the canonical orientations of trap and vcip give rise to opposite orientations on the one-dimensional connected component, such that the identity

a[u] = crIve,a] implies the relation

r(u) =

.

2. Considering the situation uf3 E M(A -= 0), vo e .A4(A = 1), we obtain at both ends the same orientation on the component from the canonical orientations of the boundary points, so that we are led to the identity

ra (uap) = 7,(vo) .

CHAPTER 4. MORSE HOMOLOGY THEORY

148

Summing up these results, we may reduce the difference

(e` —

(x,,)

from (4.15) to trajectories uo E M sc„x,(A = 0,1), which solely bound com° of type (d). ponents M x11

As to (c): Considering connected components M xH° of the diffeomorphic type (—oc, co), we now have to deal with weak convergence toward mixed broken trajectories at both ends. The natural isomorphisms Q(), to,) : Det Du, -=-% A"T( A,).A4 11 ... together with Proposition 3.14 enable us to reduce this case to the analysis in the proof of Proposition 4.5. Once again, we distinguish two types of cobordismequivalence by the order in the mixed broken trajectories: •

(ILA , iip) E M x

• Via , VA) E

(A) x .A.4f.° z o ,s0 as in the right hand side of (4.13) and

x M y.,x, (A) as in (4.14).

The associated case distinction for the equivalence of broken trajectories is illustrated within Figures 4.5 and 4.6. We observe that the difference of both

43 Figure 4.5: Equally directed induction of orientation right hand sides (4.13) and (4.14) may also be reduced to the computation for

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

149

•

Xf3

Figure 4.6: Induction of orientation in opposite directions mixed broken trajectories, which the gluing operation associates to the onedimensional components A411. ,x0 ° of type (d). Thus, it remains to accomplish the analysis of the cobordism-equivalence for components of type (d), which, due to the discussions up to this point, establish the decisive link between the left hand side and the right hand side of the chain homotopy equation (4.12).

As to (d): On the one hand, as we know from the discussion of case (a), the canonical orientation [1 0 11 induces an inward pointing orientation on the entire component M x/50 0 r-te, [0,00) at boundary trajectories uo E M, 0 (À = 0) and an outward pointing orientation at boundary trajectories u1 E = 1). On the other hand, the case distinction for the orientation induction for mixed broken )-trajectories referring to the canonical orientations and the natural isomorphism 1-2(A,u),) yields an inward pointing orientation for the order (u)„ fio) E Mrc„zo(A) x

and an outward pointing orientation for the mixed broken trajectory with the order

(fi c„ VA) E MC yc. x

(A) .

An illustration is given by the sketches in Figure 4.7. Hence, in relation with

150

CHAPTER 4. MORSE HOMOLOGY THEORY xa

uo

x.o

X/3 Figure 4.7: Cobordism-equivalence of type (d)

the boundary trajectories

uo, ul c

(A = 0,1) ,

we distinguish the following four combinations:

•

Ta ,11À) ( • T cr (-0 ) = —Ta (21.0),

•

T

•

1-,(u x ) • To. (fl, 0) = Ta (u1)

•

To,(12 a ) • Tc,(21 ),) = --Ta (Ui) •

(1-1

• To

and

We are finally able to deduce the asserted identity for (' a ,

xo)

Critkfg:

E

E

p,(z)=k+1 (A,u),)EMLI

-

E

E

7-,(u1)

Ta (U),) • To. (fl 0)

j10€1421,3(3,„

E E —E

z(y)=k -1 fiEf,4*.

u,Emx,„,,x,(1)

E

To.(fico• Tc,(2/ A)

(A,v,x)EML,

Ta(uo) •

G

Critk fa x

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

151

Summing up this last analysis, we have obtained the result on the canonical homomorphism 43),7a

11`," (f a) —)

(f )3 )

associated to given Morse functions fa and fa together with a fixed coherent orientation, by additionally taking into account the A-parametrized trajectory spaces. The next proposition establishes the final independence of the homology groups from the choice of Morse function.

Proposition 4.7 Each ordered triple of Morse functions fa , f 0 and P together with the associated homomorphisms (1)"' fulfills the composition rule

43,7° (:) Proof. At this stage we take advantage of the freedom of choosing an appropriate Morse homotopy, established by Proposition 4.6. Considering the homomorphisms (De and 4;3,, c` induced by 43:(h 13-r) and 4):(hc' 13 ), respectively, we now use the gluing construction for simply broken trajectories consisting exclusively of homotopy trajectories associated to 0-Y and PO. Due to Proposition 4.6, every Morse homotopy h(R) constructed in this way must yield the same homomorphism (Dr• But in order to be able to apply the gluing result from corollary 2.61, we are only allowed to put a finite number of critical points x,„ xo, x-y into relation with each other by isolated h-trajectories. Hence, we generally cannot find a homotopy hcrY such that the identity (4.16)

(1):(ha -1 ) • c = (43:(h° .( ) o 4)(h af3 )) • c

is satisfied for all chains c E C.(fa ). Fortunately, this is not necessary with regard to the required result for the homology groups. It is sufficient to find a homotopy hCv-Y (R(Ck)) according to the construction of the gluing for each fixed chain Ck

aixka , i E Ck(f a ), k E Z ,

so that relation (4.16) is satisfied. We consequently deduce the homological identity 43-*fa ({ckl)= { Efa(ck) • ck} = { (4)2 99 0 (I);a) (ck)} = (C,r° o 4>l,3,a) ({ckl) from Proposition 4.6 for each class

Ickl E HZ(f a ), k EZ . Thus, without loss of generality, we may start from a fixed ck = x c, E Critk fa ) so that there is only a finite number of pairs (xp, E CritkP x Critkf

with AA "" • x,„,x

x A.40 1' 4 01 "'" •xp,x7

152

CHAPTER 4. MORSE HOMOLOGY THEORY

We set the constant R = max R(s c„ xo,

according to Corollary 2.61. Thus, the identity E E E

(ti aa) • Ta (lL)

20 uo,sCA•ix,,x 0 U0.7 EMx

(4.17)

= EE so

E Ta(u.o#Ruo-,) us.y Emx,,r,

together with the bijection Mh"

za # R mx ,„Xo x mh"

from Corollary 2.61, proves the assertion. Here, the identity r(u) • r(u) = rc,(uo#Ru0) = -7-,(nory )

follows from the relation between gluing and induced orientation as already analysed above. Let us now sum up the results from the last three propositions in the second main theorem of Morse homology: Theorem 8 Given a fixed coherent orientation a on the manifold M and two arbitrary Morse functions fa and fi , there is a canonical isomorphism between the associated families of homology groups H(f)

(.0)

and the following functorial relations are fulfilled for the family f 0 Morse functions):

•

(D ZI3 o

•

4) ,C: a

fe, c` I fa ,

= idHz (fa)

.

Proof First, the isomorphism property is a consequence of these functorial relations as we see from V.13 o (1)(,3a = idm ( f.)

and vice versa. The fact that (Dr actually reproduces the identity is concluded immediately from the constant Morse homotopy f fa'

=

fa

•

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

153

Definition 4.8 Finally, this functorial concept leads us to the independence of our homology theory from the choice of a concrete Morse function. Considering the product groups

friz

H

H(r), k EZ ,

f - Morse fct.

we define H i° as the subgroup Hi° = {(. , {4'1,

, {4}, .) E 1 41{4} = ei3c a • {en, ...}

which is well-defined due to Theorem 8. This homology group, independent of the Morse functions, actually describes the same as the inverse limit lim (fa) with respect to the isomorphisms <14a . This concept will be called the identification process.

4.1.4

Topology and Coherent Orientation

Throughout the rest of this chapter we assume without loss of generality that M is a connected manifold. Since we intend to relieve our Morse homology concept from as many noncanonical assumptions and inputs as possible, the last item we have to get rid of is the coherent orientation a. Therefore, we have to analyse how the homology groups H.', change if we replace al by a-2. In a way comparable to the case of Morse homotopies, we will look for canonical isomorphisms, which enable us to identify uniquely the elements of the groups sequence { H:r }or coh. orientation} as in the inverse limit process. Actually, the key for this problem lies within the group action described in Proposition 3.16, F x CA CA Our aim is to extract homology isomorphisms from these transformations ço E

F of coherent orientations. For the sake of simplicity we first illustrate the procedure with a fixed Morse function f and consider the associated homology groups HZ(f). Obviously, we may put all coherent orientations cr E CA coinciding on the isolated trajectories of the negative gradient flow of f together in one equivalence class, that is

(4.18)

{a0}={a.ECA

([u, Du ]) = croau, DJ) for all u E x, y Crit f j •

154

CHAPTER 4. MORSE HOMOLOGY THEORY

Hence we define the set of equivalence classes as CA = CA/_. Let us now consider the group of transformations

G(f) = {±1} crit f together with the action

G(f) x CA CA, Wu,

(4.19)

(yo a-)[ux ,y ] for

cp(x)yo(y)o - kx ,v ) E .A.41,y •

kix,y} =

In fact, G(f)/± forms a subgroup of F. The next observation is that, given any such orientation transformation q, , we can indicate uniquely a suitable isomorphism of the associated homology groups 11 1 (f ) and H2 (f): Choosing

(4.20)

(VP : (C.,

a-)

—> (c„„ 84,-)

4)4, . X =- ,p(x). x,

s E Crit f ,

we compute

= a°. x =E(x , YY T Y= So(x)E(x, Y ) 6°° (,o(Y) • Y This leads to the identity

= v2(x)E(x, (4.21)

) 7 40(Y)

= z_,(x, O w°. VP • Y OW°

.

Thus, (VP represents a chain homomorphism, which is one-to-one due to the involutivity o2 = id. Consequently, each transformation yo E G(f) gives rise to a homology isomorphism. However, there is the problem that the endpoints of the [u, KI E CA merely determine the transformations yo up to the sign ±cp. This amounts to stating that the associated isomorphisms of homology are only determined up to the sign of their degree of modulus 1. Fortunately, by introducing an additional parameter, we are able to remedy this flaw within our orientation concept. At this stage we shall discuss in a more particular way the coherent orientation of equivalent operators, which coincide at their endpoints, but

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

155

whose supporting curves may be connected to an n-dimensional `Möbius band' (u • v -1 )*TM. We remember that we might put the original definition of the equivalence of such Fredholm operators along curves in a more general form, such that operators fulfilling the condition w1 ((u- v -1 )*T/11) = 0 are admitted for simultaneous orientation. Already this weaker condition, which admits more possible coherent orientations, implies the fundamental identity 0 = 0 of a boundary operator.

a a-

Example Let us consider the non-orientable manifold P2 (R) endowed with a suitable Morse function as indicated in Figure 4.8. Then, it is due to the chain

Figure 4.8: a-trajectories on P2 (R) complex property 8 2 = 0 for any arbitrary coherent orientation with respect to the weaker equivalence relation just proposed above that we compute the following relations for the characteristic signs of isolated trajectories:

(4.22)

To (Ui ) • To (V2) = — To- (U2 ) • rcr (V1) To. (U1) • To (Vi ) =

— Tcr (n2) • rcr (V2) •

These equations do not, however, determine the characteristic signs 7,(u,), 7,(7.72,) uniquely. Since u1 • u2-1 and vi • v2-1 represent Möbius bands, the

156

MORSE HOMOLOGY THEORY

CHAPTER 4.

operators

(u1 Du 1 ) ,

(7.4, r)v,)

(u2, Du2) 76 (yz, Dv2)

cannot be equivalent according to the weaker version despite the coincidence of the endpoints. In fact, we may suggest exactly two different coherent orientations up to some transformation from G(f), which yield non-isomorphic homologies, namely:

(a)

(Hg',

, Hg' )

(b)

(Hg 2 , H2

,

H2)

(Z, Z2, 0) (

z

,

and

z)

If we make use of the original equivalence relation, however, which orients the 'Möbius bands' in a unique way, exactly one of these two possibilities is selected. Referring to this example, we observe that possibility (b) violates the known identity for the homology of connected manifolds, Ho (M) Z, which is principally related to the dimension axiom. Actually, case (b) yields nothing more than the classical cohomology of P2 (R) turned around by reflection, that is (H2 , HI , HO) (Z2, Z) This means that, already at this stage, we find a hint of Poincaré duality, which is immanent to Morse homology. It is closely connected to the choice between positive or negative gradient flow for the definition of boundary operator. As to the example, in order to comply with the dimension axiom, we have to decide on case (a). As we shall see in Lemma 4.13, this is related to the condition of the equivalence relation for operators in Definition 3.7. That is the condition that we simply demand the coincidence of the trivializations at the negative end, provided that we start from the Morse homology associated to negative gradient flow. Otherwise, choosing the positive end for the coincidence condition would lead us to the cohomology turned around by reflection.

Definition 4.9 At first we define an appropriate generalization of G(f) with respect to arbitrary Morse functions. Hence, we have to consider the set of

Hessians E=

Am E End(TM)

: Tti,M Tm M non-deg. and conj. self-adjoint

instead of Crit f. Let A = { ±1 }E

be a group with respect to pointwise multiplication together with the action A X CA -> CA

(4.23)

cr) (,ocr)

(Per

K;,

= so(K;)(,0(K:)a[ux ,y , K; , K] .

4.1.

THE MAIN THEOREMS OF MORSE HOMOLOGY

This yields a generalization of { ±1 }Crit Crit

f

f

157

in the sense of

E H2 f (x) .

x

Once again by analogy with (4.20) and (4.21) the yo E A give rise to isomorphisms (4.24) : (c.(f), a-) (c,,(f), which additionally are compatible with the isomorphisms

Ilf: (fa) =%-

(f0 )

on the level of the homology groups. Hence, the problem indicated above may be expressed as follows: A =F i± ; that is, there are exactly two transformations ya E A associated to a given pair (ai.• az) of coherent orientations, such that the identity az = (pal is satisfied, namely ±c,o. In order to attain a unique identification of the homology groups by means of the isomorphisms v. = (1)w associated to the 99 E A, we crucially require the functorial behaviour (4.25)

So

= ça. ° So. •

This condition prevents us from choosing erratically between -1- fp and —(p. Actually, the problem of a natural selection process with respect to functorial behaviour lies in the relative constitution of a coherent orientation and its transformations. These transformations are given by ça(x)ça(y) with respect to the pairs (x, y) of endpoints. Thus, the relative character of these transformations is in contrast to the absolute character of a mapping (lc) : E {±1}. In order to fix such a mapping cp uniquely, we need a characterization of (,c) by Fredholm operators, which are endowed with only one end within E. This means that, instead of merely orienting the relative, connecting trajectories from Mx,y = WIL (X) Ws(y), we have to find an absolute fixing, for example by means of an orientation of some stable manifold Ws (y). Thus we shall develop the following concept of the so-called 'one-sided' operators to find a solution for this problem. Let us first regard the trivial case, M = R. Let A+ G S be a conjugated self-adjoint operator on Rn and A E C00([0, 00], End (n, R)) endowed with A(+oo) = At . In this situation, we consider [0, co] c to bear the differentiable structure as a submanifold with boundary within R. We consequently define the one-sided operator (4.26)

SA : X = H12 ([0, 00), Rn) u 1—>

y = L2 ([0 , 04 Rn) +A•u.

158

CHAPTER 4. MORSE HOMOLOGY THEORY

Referring to the methods which were developed throughout the Fredholm Section 2.2.1, it is straightforward to show that SA represents a Fredholm operator with index n tz(A+). We may conclude similarly that

(4.27)

0(A+) =- { SA E .F(X; Y) I A (+ oo) = A+ 1

is a contractible space such that we are able to equip the determinant bundle Det on e(A) with a well-defined orientation. This concept of a one-sided operator in the trivial framework can be transferred to the manifold M as follows. Ax E

(4.28)

Let s E C c° ([O, co), M) have the fixed end s(oo) = x and let E be fixed. Then we consider the pair (s, SA), where S : H 1,2 (s * T

[0,00)) —> L 2 (S f T MI [0,co)) U

Vt u+A•u

is a Fredholm operator of type (4.26), endowed with the fixed end A(+œ) = A. This set {(s, SA)} of one-sided operators with fixed end Ax may be assembled in a uniformly orientable equivalence class [SA] . The uniform orientability is due to the fact that, by analogy to Definition 3.10, the choice of orientation of an arbitrary representative (s, SA) together with a trivialization of s*TM induces an orientation of 0(Ax+ triv ). Hence in a reverse and unique manner, this induces orientations for all the other representatives, so that they do not depend on the actual choice of trivializations. This may be concluded by means of a lemma which is strictly analogous to Lemma 3.8. Moreover, it proves to be consistent with the fact that the space of all curves s with a fixed end s(oo) = x is contractible to the constant curve s H x. Additionally, we are able to analyse a gluing operation for such onesided operators together with the two-sided' operators (u, K) — u *Tm of the former type: [SA.]#[u, K] [SR-, ] for Ar = K_ . Starting from a given coherent orientation with respect to the classes of operators [u, K ] E A, we derive a unique orientation of [Sic + ] from a given orientation of [SA.] by means of this gluing operation. In order to obtain uniqueness guaranteed, we obviously need independence from the chosen curve u associated to the fixed endpoints. This is the very point where we remark the necessity of a simultaneous orientability of all operators (u, K) belonging to given ends Ks K Let us state the consequence of this uniqueness as

Proposition 4.10 A coherent orientation of the relative classes of operators5 [K;, Kin together with an orientation of a given fixed [SAJ, A

E

E induce

5with respect to the equivalence of orientations by means of trivializations with coincidence at the negative end

4.1.

THE MAIN THEOREMS OF MORSE HOMOLOGY

159

unique orientations of all positively one sided classes of operators [510 ],K -

-

E

e. On the other hand, however, let us regard what we have to change, if we wish to endow the classes (S K -1, 1Ç- E E of negative half-sided operators with a uniform orientation, that is in principle an orientation of the unstable manifolds. In the case of a non-orientable manifold M we would also have to change the definition of equivalence of the relative operators (u, K) v, L). Namely, we would have to choose the coincidence of the trivializations of u*TM and v*TM at the positive end u(-1-oo) = v(+co) as far as 'n-dimensional Möbius bands' w1((u.v -1 )*TM) 0 are concerned. Actually, the question of whether we have to endow either the 'stable' or the 'unstable' manifolds with a uniform orientation corresponds to the decision between negative and positive It is exactly one of the two alternatives, namely the choice of gradient flow. the stable manifolds in the case of the negative gradient flow, which reproduces the necessary identity for the classical homology of connected manifolds (see below), Ho(M) Z, also in the situation of a non-orientable manifold. Proof of proposition 4.10. Without loss of generality, let s E C'([0,cx)], M) and ul, u2 E C°°(R, M) be asymptotically constant and satisfy the identities s(+oo) = u l (-oo) = 1L2(-co) = x and ui(+co) = u2(+0o) = Moreover, let us assume a positive half-sided operator Sz on s together with relative operators K1, K2 on u 1 and u 2 , respectively, once again asymptotically constant with = . = = We consequently consider asymptotically constant trivializations çbx : s* T M

[0, oc] x Rn,

q5 1 : uTTM

R x Rn ,

u;TM

>118 x Rn

satisfying 0,(+co) = q51(-oo) = 02( -oo), such that the pair (1,2) is admissible and the non-degenerate, conjugated self-adjoint endpoint operators 6 A, = 01 (S1 )(+oo) = 01,2(1(1,2)( — 00), Ay

=

q51,2(K1,2)(+oo) E GL(n,R)

are diagonal matrices. If the identity çb i (±Do) = ¢.2(±oo) holds at both ends, the proof is straightforward. Thus, let us assume the actually interesting case 60. (Sx) = Ox 0 So 0; 1 E .C(X; Y), see 4.26

CHAPTER 4. MORSE HOMOLOGY THEORY

160

01 (+CO) = So 02 (+C), where S denotes the reflection matrix

+1

s=

from Definition 3.7. Let us henceforth use the following notations:

= 0s, = 1, = 2,

= S ° 0x, = S 0 01, = 02

0; 0102'

.

We compute (0; # ° 0 -E ) (+co) = (0-;#°0 -1E+ ) (+co). Now let [Sx ] be oriented firmly by o(S) as well as K1 K2 by o(Ki) and o(K2). Then, the relation between the orientations

(4.29)

o(Sx #°K1) o(S)#o(K i )

o(S4#0(K2) 0(Sx# ° K2)

amounts to relating

(4.30)

(0; #° 02-± ) (0(Sx# 0 K2))

( 4 #°4)

on the contractible class 8(Ay ) in trivialized form. Thus, we have to verify that this relation is equivalent to the compatibility of the orientations o(Ki) 0(K2) according to definition 3.10, that is

(4.31)

0-11-+ (0(Ki))

0-21- (0(K2))

in

e(Ax , Ay ) C Etriv

since (4k,, 4-) is admissible. 7 We notice that we may state relation (4.30) in the trivialized case alternatively as

(4.32)

4

(o(sx ))#4+ (o (K l )) a, 4); (49(Sx))#02 ± (0(1(2))

Now let Lt be any homotopy between 0r(K1) and 02 + (K2) in the contractible space e(Ax, Au ). Given a large enough gluing parameter, it induces a homotopy 0:(Sx) #Lt between 4 (Sx)it0i + (KO and (Sx )#02-± (1(2)Then the identity

(0 (Sx)) = (S ° 0; ) ( 0(Sx)) = - 0; (0(Sx)) together with the equivalence of orientations (4.32) imply the relation

(4.33)

4+ (o(K1))

7See Definition 3.7.

-02-+ (o(K2)) = 0 2-F- (o(K2)) .

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

161

Before we are able to relieve the homology groups HZ (M) from their parameter a, we first have to introduce even an additional parameter. Choosing any class [Sit] , A e E, of half-sided operators with fixed positive end and specifying any orientation o(SA) of this class, we expand the parameter dependence,

H° •-•-■ H'°(sA)) . For instance, if we choose the Hessian A = H2 1 (x), x E Crit f for a given Morse function f, specifying the orientation o(SA) amounts in principle to fixing an orientation of the stable manifold IV (s) . Hence, the coherent orientation a gives rise particularly to an orientation of connecting orbits ux ,v E v= W u (x) MIVs (y) thus leading to orientations of all the other stable manifolds. Now, let (a, o(S A)) and (o-',o'(SA' )) be pairs of parameters, such that there is exactly one transformation pair ±yo (see above). It is due to the coherence of the given orientations and to Proposition 4.10 that we may start without loss of generality from identical classes [SAI = [S'A ] . Let us choose the sign of ±cp according to the case distinction (4.34)

ço(A) =

+1 —1,

if o(SA) = 0/ (SA) if o(SA) = +(ASA) •

To put it in other terms, we extend the action ±1 1 E x C A CA

to (p • (a- , o(S A)) = ((p o - cp(A) o(SA))

,

so that we deduce the existence of a now unique transformation map ço E { ±1 le between (a, o(S A)) and (u', (S'A )) from the coherence condition for a- and o-'. At last, due to the construction, this mapping ((o-,o(SA)),

oi(S/A )))

satisfies the functorial property (4.35)

(pa'acr2o (i)cr2cri. = (p° 3 ' 1 ,

= id ,

which was demanded in (4.25). give rise to isomorphical homologies, which may be identified with each other by means of the isomorphisms in a unique way due to (4.35). As a consequence, this identification process provides us with homology groups Hk(M) which are finally independent of any other parameters than the order k and the smooth manifold M. Thus, coherent orientations

CHAPTER 4. 'MORSE HOMOLOGY THEORY

162

Definition 4.11 By analogy with definition 4.8 we define the Morse homology groups as the inverse limit

H'°

(M) = lim (M)

with respect to the isomorphisms Referring back to definition 3.7, our argument for the construction of the non-trivialized Fredholm-classes on the one hand was founded on the relation for the halfsided operators as described in proposition 4.10 and on the Z for connected manother hand on the algebro-topological relation Ho (M) ifolds. The discussion of the latter, which is equivalent to the dimension axiom within the axiomatic framework is now anticipated to the next section. We verify the consistence of our orientation construction with this crucial relation as follows.

Proposition 4.12 Let M be a connected, smooth manifold equipped with a

Morse function f, which possesses exactly one local minzinum y o . Then, the Morse homology group of order 0 with respect to f satisfies the relation H0(f) -L2 Z The proof is based on the following fact: Lemma 4 13 Given two different, isolated trajectories ui and 112 from .1+.41 ,y E Critif x Crit of,, that is with(x,y) (ui, Du i

(u2, Du2 ) ,

any coherent orientation a leads to the identity 7-0.(ui) = --Ter(u2) • Proof We consider the connection of the curves at the point x referring to a suitable reparametrization which yields

-y =

u2 E Cc° ([-1, 1],M) .

This is nothing else than a suitable parametrization of the 1-dimensional unstable manifold Wu (x) as a 1-cell. We now choose a trivialization of -y*T M, such that on the one hand the Hessians H2 f (x) = Du- = Du-, and H2 f (y) = Du+, = DI, become diagonal matrices with respect to this trivialization, and such that Rn differ on the other hand the induced isomorphisms TM = T.7(±1 )M by the reflection S defined above if they differ at all. Hence, this trivialization

THE EILENBERG-STEENROD AXIOMS

4.2.

163

gives rise to an admissible pair with respect to (ui , D. 1 ) — (u2, D 2 ). Moreover, we may construct this trivialization in accordance with the splitting-off of a one-dimensional subbundle n of = -y*TM,

Here, n denotes the one-dimensional bundle on [-1, 11 induced by TWu(x) with the fibre ljrj = Wu(x) which is identical with the eigenspace of H2 f(x) associated to the unique negative eigenvalue. Up to an appropriate parametrization, the generating sections fib 212 for the kernels of D ul and Du2 , respectively, lie exactly in this subbundle n Thus, starting from a fixed orientation on (u -1 , we can derive orientations of the bundle = (u -11 • u2)*TM from the canonical orientations [U1 ] and [fi2 ], which must be opposite each other. This proves the lemma. .

Proof of the proposition. The unique local minimum yo of the Morse function f provides us with a generator of the cycle group Zo(f) for the Morse complex of f. This generator is unique up to sign. We deduce from the lemma that, given any coherent orientation a, the boundary 8°x 1 of each generator x 1 E CI (f ) vanishes. Thus, the assertion follows from Bo( f) = {0}. It seems natural to expect that the proposition remains true if we do not impose the special condition on f to possess only one local minimum. The proof in this general setting requires some more steps and will be given in the next section in relation to the dimension axiom.

4.2

The Eilenberg-Steenrod Axioms

In this section, we shall carry out the final step of the development of Morse homology theory. We shall verify the functorial properties and the accordance with the axioms of a homology theory as they were set up by Eilenberg and Steenrod in [E-S ], namely, • the existence of a long exact homology sequence, • the homotopy axiom, • the excision axiom and • the dimension axiom. For the sake of simplicity, we shall first derive the functorial construction and homotopy invariance for the concept of the absolute homology groups H.morse( ) before we treat the somewhat more involved relative groups Iism °r" (M, A) associated to admissible pairs of manifolds (M, A). In this framework

CHAPTER 4. MORSE HOMOLOGY THEORY

164

a natural induction of chain maps for Morse complexes by closed embeddings of manifolds proves essential for functorial construction. In order to discuss this construction, however, we first need some technical preparation, namely suitable extension lemmata for Morse functions. 8

4.2.1

Extension of Morse Functions and Induced Morse Functions on Vector Bundles

The problem of the extension of Morse functions from submanifolds of M to the whole manifold M may be separated into a technical one on the one hand, the extension of the coercivity property and of the regularity of Morse functions on open submanifolds, and on the other hand, a problem related to dimension with respect to closed submanifolds. The first lemma yields the extension of the coercivity property. We notice that the coercivity of a continuous function f : M —> R, i.e. .. a } /VP = { x E MI f(s) ., is equivalent to

compact for all a E R ,

the following property:

a constant c> 0 such that the continuous function fc = f + c : M —> [0, oo) is non-negative and proper, i.e. f 1 (K) is compact in M for all compact K C [0, oo) . There is

Lemma 4.14 Let M be a smooth manifold, A c M be a closed subset and f : A —4 [0, oo) be continuous and proper. Then there is a continuous and proper extension

g:M —>[0 , 00 ) gIA= f ' Proof. The proof is basically founded theorem for continuous functions:

on an application of Tietze's extension

Let X be a normal topological space, A c X be a closed subset and f : A —> [0,1] be a continuous function. Then there is a continuous extension f* : X —> [0,11, ri A = f . Since we are able to embed M as a closed submanifold in an RN with N chosen suitably large, we may start from the assumption that M = RN without loss of generality. Denoting the closed ball with radius r by

13, = {x e RN I 8for coercive Morse functions!

11x11

.... r}, r > 0

4.2. THE EILENBERG-STEENROD AXIOMS we define (4.36)

165

fn—f lAnBn

Consequently, there is an increasing sequence of positive numbers R+ , such that the condition

(rn)nEN C

fn (A n fin ) C [0 , r„] is true for all n E N. This enables us to apply Tietze's theorem inductively for n: For each n E N there is a

(4.37)

: no

> 1 0, r,,] satisfying

—

I A n B,

= fn.

and

g;5;1Bn- = g' ?,-1 •

Defining

hn = arctan

: Bn —> [0,

nGN,

we obtain a sequence of continuous functions once again according to Tietze's theorem,

k,„ : RN —■ [0, —7 -; 1

with

kni Bn = hn and lislii n2cokn =

71"

.

It is due to the construction in (4.36) and (4.37) that we may state the uniform convergence on compact sets,

(4.38)

kn, C14 k E

(R N , [0, 12- ) )

with

lim k(x) = 11x1Hcc

This implies a continuous and proper extension of

it

f by

g = tan k E C° (RN , [0, oo)), gi A = f . D

We now have to transfer the extension result to the dense subset

Ce*(M, R) C C° (M, R) of the smooth functions. Furthermore, we have to add the regularity condition

df ih M C T* M . We refer essentially to the results of differential topology as they can be found in [Hi]. Hence, we take over the notations C(M, R), Ar(f, c), etc., for the strong or, equivalently, the Whitney topology and the basis sets on Cg.(M, R)

.Ar(f, , e) = { gJ ig(x) — f (s)I < e(x) for all x E M }, CE C° (M, R+ ) . An analogous definition holds for Cig(M, R). Then the extension result for smooth Morse functions can be stated as

CHAPTER 4. MORSE HOMOLOGY THEORY

166

Lemma 4.15 Let M be a smooth manifold, let A be a closed and W an open subset and let f E Coe (W, R) be a Morse function on the submanifold W, such that Critf CACAcW cM Then there is a smooth Morse function g on M which extends f, that is, glA = f1A . Proof. Since M is a normal topological space, we find an open subset U satisfying

C

M

AcUcTicw Due

to Lemma 4.14 there is a continuous coercive extension

(4.39)

go M

901u

f -u •

The set of continuous and proper functions represents an open subset with respect to the Whitney topology (see RED, Prop°s (M, R)

C C(M,

R), open subset .

Therefore, we find a convex neighbourhood (4.40)

.A1(g0) C Prop°s (M, R) C CI (M, R) ,

which contains merely coercive functions. Furthermore, referring to the C2Whitney topology, we can find a convex, open neighbourhood .A7i.(go) -

(4.41)

C Cl(M,R)

with

Crit n U\A = 0 for all it

E

.

Thus, the intersection N. = n At., is an open and convex subset of C(M, R), too. It is clear from the well-known transversality theorems in differential topology (see [Hi]), that X=ffEC 2 (M, R)Icif (11M cT*M1

forms an open and dense subset of the Baire space C2s (M, R). Since C"(M, R) lies dense in C3 (M, R), too, there is a g' E Cc° (M, R) nArriX

Now let a M ---+ [0,

1

]

arbitrarily close to go in C3 (U\A,R) .

be a smooth function satisfying and

167

4.2. THE EILENBERG-STEENROD AXIOMS

We consequently define (4.42)

go + a • g' .

g -= (1 —

Due to the concrete definition of the convex neighbourhoods identity of go and f when restricted to U, g satisfies the properties

geCœ (M,R)nArnX

and

and the

giA=f1A

provided that giu\A has been chosen close enough to go u\A• As pointed out above, the next step is to develop an extension result for closed submanifolds. Here, we require an additional construction method for Morse functions, which appears very naturally. A feature which proves crucial for the whole Morse homology is the canonical tensorial behaviour of Morse complexes with respect to product operations: Given two manifolds equipped with Morse functions (M, f) and (N, g), the operation (4.43)

(f g)(rn, n) f(m) + g(n)

naturally provides a Morse function f ED g E Cc° (M x N, R)

together with the canonical identification (4.44)

Critk (f ED g) = U Critif x Crit 3 g EMxN , i+j=k

because the Morse index behaves additively with respect to the operation 0. At this stage, regarding our current concern, we need a generalization toward smooth vector bundles :EM. Let (., .)E be a Riemannian metric on E with the associated quadratic form

q(t)m ) = ( and let f be an arbitrary Morse function on M. Then we obtain the Morse

function (4.45) fE(vrn) = f (m) + q(vm) on E with the property

(4.46)

fEi M = f and

Crit.fE Crit. f .

168

CHAPTER 4. MORSE HOMOLOGY THEORY

On the one hand, the latter is due to the local product structure of E, so that the results (4.43) and (4.44) may be transferred to this special bundle situation. On the other hand, we observe that q restricted to the fibers is a trivial Morse function, that is Crit.qm = Critoqm = {O m } . This feature enables us to derive the following extension result:

Proposition 4.16 Given a closed subrnanifold A c M and a Morse function f A E COE'(A, R) on A, there is a Morse function f E Cœ) (M, R) extending

f A, fA fA = Proof. Having in mind the above preparations for naturally induced Morse functions on smooth vector bundles, we consider a tubular neighbourhood W of A, which always exists for any closed submanifold (without boundary):

to,

Wc M

:

A I' is the normal bundle of A and W is an open submanifold. Thus, the diffeomorphism cp induces a Morse function on W,

Here,

(4.47)

fw = f o‘,0 -1

which extends fA : fw i A = fA . The preconditions of Lemma 4.15 are fulfilled if we choose a suitable restriction: Let R> 0 and AR = ça ( C 1 ([0, R])), that is, AR C AR+E CWCM, AR closed . Then, Lemma 4.15 gives rise to a Morse function f on M satisfying f A R±E that is, in particular,

= fW I AR±e

11A = fA•

Actually, we may gain a further important result from this special extension construction on closed submanifolds: Corollary 4.17 The extension f of f A in the above proposition can be chosen in such a way that there are no trajectories for the negative gradient flow off leaving A.

4.2. THE EILENBERG-STEENROD AXIOMS

169

Proof It is due to construction by means of quadratic extension on the normal bundle u that the negative gradient field —V f, represents an inner normal vector field on the boundary a gv-1 ([0, e]) of the manifold with boundary qz,-1 ([0, e]) for any e > 0. This means that —V fw is an inner normal vector field for (9 A6 in A,. From this we obtain the corollary.

4.2.2

The Homology Functor and Homotopy Invariance

The DifFeomorphical Functor First, we wish to discuss the homomorphisms between the respective homology groups induced by diffeomorphisms. We assume

(I) : M

N

to be a smooth diffeomorphism between two manifolds and f to be a Morse function on M. Obviously, (p gives rise to the Morse function

(4.48)

,of= f.99'

on N, so that we obtain the one-to-one correspondence

(4.49)

99 : Crit f

Crit,(99,f) .

However, before we are led to a chain map between the associated Morse complexes (09„ : C.( f) f) we have to verify the condition

(4.50)

So.

(P. 3 •

Provided that the generic Riemannian metrics on M and N have already been fixed, it is generally merely an isometry cp which maps isolated trajectories for f onto isolated trajectories for v.,. f . But we are able to choose the generic metric g on M in a way that also provides a generic metric on N via induction by (p. This means that we may choose the metrics generically such that (p becomes isometrical. Remember that the transformations of the Morse complexes associated to fixed Morse functions by means of suitable Morse homotopies. Analogously, we may regard the homology groups as independent of the Riemannian metric and we therefore assume without loss of generality that the given 9, is isometrical. The only item left to verify with respect to (4.50) is the consistency of the coherent orientations which have been chosen for M and N independent of cp. This can be achieved by means of an appropriate

170

CHAPTER 4. MORSE HOMOLOGY THEORY

transformation by an a e {±i}cr"f as we described in the last section, that is

(4.51)

C(f) C.(V.f) (P.xk = a(xk) • (p(xk), xk E Crit k f .

To sum up, up to a respectively appropriate homotopy of the Riemannian metric each diffeomorphism :M N induces a homology isomorphism So+ 11:m (M, f)

Hf.' N (N, So. f) •

Now, regarding a homotopy of Morse functions f.

f 13

,

we can confirm functorial consistency. ço induces a homotopy cp* fa yo4lt

cp* pp

such that in the case of an isometry the h-trajectories associated to

Vht

=

,y

+ litt1 2 1Vhd 2 are mapped by yo onto the corresponding h-trajectories belonging to ço. ht = ht o yo-1 . In a nutshell, the equality

(4.52)

4'. (42.P, (Pin 0 (P. = (P. 0 (1).(f° , r)

implies that ço in fact induces a homomorphism

ço* : H(M) — > H(N) according to the definition of 11* (M) = lim H* (M, fa). Obviously, the construction guarantees consistency with the compositions of the morphisms

(4.53)

(tp

o 0) * = (p„, o .0* ,

so that we obtain a functor as far as smooth manifolds together with diffeomorphisms are concerned.

The Embedding Functor and the Homotopy Lemma By means of the extension results and this diffeomorphism functor we are now able to accomplish the extension to closed embeddings. Let ço : M 'N be an embedding such that v(M) represents a closed submanifold within

4.2.

THE EILENBERG-STEENROD AXIOMS

171

N. We henceforth denote by ço°,, f the Morse function on w(M) which is induced by f and w and which can be extended to a Morse function yo. f on the whole manifold N without any negative gradient flow trajectory leaving the submanifold w(M) according to the corollary of Proposition 4.16. Actually, this condition allows us to identify the Morse complex C. ((e. f) on cp(M) with a subcomplex of C. (.f) on N, that is aso- f

(4.54)

_

I c.(gf) -

.

Here we assume a suitable extension of the coherent orientation of w(M) to the whole N. Thus, the identification with the subcomplex together with the diffeomorphical functor with respect to the diffeomorphism

w

:

M

. - -'' ■ço(M)

gives rise to the chain map Cs.(M, f) —> C.,(N, v. f) and consequently to the

homomorphisms cp.

:

H.,,(M, f) — ■ II.,(N, ço, f)

and ço. : H„(M) —> H,,(N) with respect to the identification process. Once again, we verify the composition rule, so that we obtain the sequence of functors

(4.55)

Hk :

( manifolds, closed embeddeddings ) —> AB, 9 k

E No .

Before the next step of the generalization aiming at arbitrary smooth mappings, we first prove the crucial Lemma 4.18 (homotopy lemma) A smooth 1-parameter family of closed embeddings (ç c't ' M '—' MtE[o,i] implies the identity

The proof of this lemma can be reduced to the following special case: Auxiliary Proposition 4.19 Let f be a Morse function on M and i9 , il : M , — ■ IIk x M be the embeddings given by i' (rn) = (v, 9category of Abelian groups

rn) for v = 0 , 1 .

172

4. MORSE HOMOLOGY THEORY

CHAPTER

Additionally, we consider the quadratic functions

q,, : R R q,(x) = (x — v) 2 , v = 0, 1 . Then the isomorphisms

: H.(f)

f)

induced by the canonical identification Crit.f homology isomorphism :

11.(qo f)

Crit. (q,,e1f) and the canonical

11,„(q1

f)

fit in with the commutative diagram Ha (qo +

H.(f)

Conclusion: If we carry out the identification of the homology groups for Morse functions f on R x M, we obtain the identity

io. _

(4.56)

H.(m)

H.(1)

H(R x M)

from this commutative diagram. Proof of the auxiliary proposition 4.19: We have to verify the existence of a

Morse homotopy qo f qi f , Ck (qi e f) map such that the associated homomorphisms (11 kh t : Ck(qo f) the generating critical points to each other according to the formula

(4.57)

h

4) k t (0 Yk) = (1) yk) 1

for all Ilk E Critkf, k E N . (R, [0,1]) satisfying

We consider a smooth, monotone function a E

10,

,

a(t)

=

t

1, t

0

4.2.

THE EILENBERG-STEENROD AXIOMS

173

and we define the homotopy tE[0,1]

(4.58)

— a(t)) 2 + f (y) .

ht(x, y) =

It is clear that this 'translation' represents a Morse homotopy between qo e f and qi ED f. For the sake of simplicity we content ourselves with a consideration of the homomorphism 4>kh t with respect to the isolated h-trajectories for the time-dependent negative gradient flow

OW,

(4.59)

'5(t)) = —Vitt o

(OM,

ry(t)) . -1

We leave out the norming by the scaling factor (1 + iht121Vht12) . This is admissible for the following reasons: First, we will derive explicitly the necessary compactness results for this special Morse homotopy. This feature was the only reason in general for the scaling. Second, by means of a homotopy using an additional parameter A, we may verify that the homomorphism (I)k on the level of homology is independent of the special form of this norming. The isolated h-trajectories

(0, -y) c Mt zi , z =

(v, yuk ) E Critk(q, e f), u = 0,1

are determined uniquely by (4.59) as solutions of the differential equation (4.60)

/6.(t), ;y(t) ) = (2 (a(t) —

OW),

—V f o

Now let if be the isomorphism

a

E

Etriv •

Since 6 is a smooth function with compact support according to the construction, we observe that

E C00° (R, R) C L2 (R,

.

As a consequence, tlf -1 (6) is determined uniquely. Thus, Cc° (R, R) solves the equation (4.61)

+ 2)

= W(a) — 6 = 2a .

To put it in simpler terms, )30 is the unique solution of ijo(t) = 2 (a(t) — (30(t)), t E ; (30 E

00 =

—

(a) E

174

CHAPTER 4. MORSE HOMOLOGY THEORY

This proves the statement that each isolated trajectory Po, E M zlipt corresponds to exactly one trajectory of the time-independent negative gradient flow for f. This amounts to the one-to-one correspondence )14]f

• x,y

`Zo,Zi

-Y

(00, 7)

with zo = (0, x) und z1 = (1, y). Due to the assumption we are considering the relative Morse index p(x) = p(z o ) = p(z1 ) = p(y). Thus, the timeindependent f-trajectories in question have to be constant, such that M zhot, zi of the trajectories consitexaly zhO,zi =

(00, Yk ) E l'z12,z1(R x M)

yk c Critkf }

From this we easily conclude the assertion.

Proof of the hornotopy lemma. Let us consider the commutative diagram of closed embeddings for any t E [0, 1]:

c

(Pt

.t zm . R x M ED

N

Rx N

where 0(0 = (t, v t (x)) ,

i 1 (x) = (t, x), i(y) = (t, y) • Since we are already allowed to deal with the covariant homological functor for closed smooth embeddings, we deduce the identity (4.62)

o it= i 0 (pt . : H(M)

x N) .

The auxiliary proposition states that isomorphisms i tm . and i pend on the parameter t. Hence we obtain the identity (Pt*

independent of t c [0,1], also.

do not de-

= ils1,17 1 ° 0* ° iM*

I=1

4.2. THE EILENBERG-STEENROD

AXIOMS

175

The Homology Functor for Arbitrary Smooth Mappings In the next passages we concentrate on projection mappings of the type p:Mx N N , which play an essential role second to that of closed embeddings. The idea is to extract the functoria1 behaviour of the homology morphisms step by step from the projections p k : R k x M M. By analogy with the 1-dimensional case in the auxiliary Proposition 4.19 we consider the canonical identification Criti (f)

Criti (qk ED f), j = 0, . , n,

with respect to the canonical quadratic form

qk (x) = (x, x), x E R k . First, we again obtain the isomorphisms i!`, : 1/* (f) : H+ (M) -Lo H * (

H* (qk ED f) yielding

M)

independent of the Morse function f. This is due to compatibility with Morse homotopies. Thus, the inverse mapping by means of the identification of critical points represents the homomorphism p! induced by the projection map p k ,

(it)

H.(R k x Al)

ik+1,k : Rk x

Rk+I x

(4.63)

H(M)

Using the notation

• Xk),

In)

1-4

((z1 • • •

xk (3 7 • • 1

O

),

m)

we may immediately set up the commutative diagram

H(M) •k 7*

ED (4.64)

1*

H* (Rk x M)

14+ ,k

o H* (Rk+1 x M)

k+I

P*

P*

H(M) which describes the relations between the inclusions and the projections as far as the trivial extensions by Rv are concerned.

CHAPTER 4. MORSE HOMOLOGY THEORY

176

Definition 4.20 Resorting to a suitable closed embedding çok : M c-0 Rk we

are able to factor the projection map p:MxN —* N into p = p k 0 ( (p k x id N ) . According to this factoring we define 11(M x N) —, H(N) y idN). .. P* = P! 0 ( , kx Obviously, we have to verify that this homomorphism A. does not depend on the noncanonical choice of the embedding çok. For this item we rely on the

homotopy lemma. Let 01 : M , — R1 be another closed embedding Extending both given embeddings by means of the inclusions i k and i 11 , i.e. ik-I-1,k 0 wic

and

ik+I ' l 0 01 : M ‘—

and regarding the functorial behaviour of the homological morphisms with respect to closed embeddings together with relation (4.64), we may start from Rn without loss of generality. Due two closed embeddings cp, 0 : M to the homotopy lemma and the uniform product structure it is sufficient to construct a homotopy ço ,-, tp. Since we are already allowed to exploit the functorial behaviour with respect to the composition pn , 2 n 0 gtz,tz , 41., we are free to compose embeddings ço and ti) with the inclusion i 2n,n : Rn ,__,

Summing up, it is sufficient to find a homotopy

(4.65)

yo t tp, Ot : M c -0 R2" closed embedding for all t E [0, 1] .

Let T : [0,1 ] ■ [0,1] be a smooth monotone mapping, locally constant at 0 and 1 satisfying T(0) = 0 und T(1) =- 1. Then we define the homotopies —

01 : Mx [0, 1] —, Rn ED Rn , Ol (. , t) = ( (1 — r(t)) • (p, r(t) • (p)

and

02 :M x [0,1] _, Rn (s Rn , 02(-,

t)

-

(r(t) • 0, (1- 7-(0)

• ,}0) •

Thus we obtain smooth homotopies of closed embeddings for each fixed parameter t satisfying

(9;,, 0) '4 (0 , (p) 1, (0, 0 .

4.2. THE EILENBERG-STEENROD AXIOMS We finally

177

reparametrize this composition of homotopies obtaining Ot = {

4t, 0 ‹ t 1 2t—i, 1 --,-- - ----- -

,/,2 w

as a smooth homotopy due to the definition of r . Obviously, required properties.

Ot fulfills the

By means of this method of factoring through a closed embedding cp : M ,— Rk we may derive the identity

(4.66)

p. 0 ic . = idx,„(N)

for arbitrary inclusions ic : 1■1 ,—, M x N, y from the

i—■

(c, y), c E M fixed,

homotopy lemma.

Now we are able to associate to any arbitrary smooth mapping between manifolds a homomorphism between the respective Morse homology groups. Definition 4.21 Let f E C°° (M, N) be a smooth mapping. Then the graph-

embedding (id, f):1l1 ,—MxN represents a closed embedding, so that (id, f). : 11.„(M) —> H,,,(M x N) is well-defined. Resorting to the projection homomorphism constructed above,

p. : H,.(M x N) H.(N), we now define (4.67)

f. = p. o (id, f). : H.(M) — ■ 11,.(N) .

To sum up the functorial construction, we choose indirect way toward the general homological functor via the factorization f = p o (id, f) into mappings which can be treated more easily. We now have to verify that this definition is compatible with the already existing functor for closed embeddings and projection maps and that the cofunctor property of a covariant functor

(g o f), , , = g„,c) f,, is satisfied.

178

CHAPTER 4. MORSE HOMOLOGY THEORY

Auxiliary Proposition 4.22 Given a closed embedding f : M directly induced homomorphism (cp f). from (4.55) is identical to

N, the

Is = (99f). : H.(M) —> H5 (N) . Proof. Let (pm : M Rm be a closed embedding. We consequently obtain the commutative diagram

id

The two closed embeddings

00 = i o f and

01 ((p x id) o (id, f)

can be related by means of a homotopy through closed embeddings qSt = (t ço, f) : M

Fen x N, t E [0,1] .

Then the homotopy lemma implies the identity (too. = 01 a from which the El proof follows immediately due to the equality p. o = id,. We are able to prove analogously Auxiliary Proposition 4.23 The projection map p:M x N —) N directly gives rise to the homomorphism ((Pp)* from Definition 4.20, and this homomorphism is identical to p = (c,op ). : H.(M x N) —)11.(N) . The last step toward the covariant homological functor follows from Auxiliary Proposition 4.24 Any two smooth mappings and g G C°9 (N, P) satisfy the composition rule (g o f),

g. o

.

f G C(M, N)

179

4.2. THE EILENBERG-STEENROD AXIOMS Proof At first, let us consider the commutative diagram

, JO, id)

xN

(id,g)

=2

(4.69)

RmxNxP

(id, g)

P3

P2

NxP

P.

Due to the definition of i., p., etc., the following identity is true: P2* 0 (id, g) = p3 ,0 (id x (id, g)), 0 it .

(4.70)

N, we obtain the isomorphism As to the projection map A : Rm x N 11.(Rm x N) and hence the identity = (A.) 1 : H.(N) (4.71)

9 , 074, = 3 , 0 (id x (id, g))„ : H.(Rm x N)

H.(P)

from (4.70) according to the definition g. = p2. 0 (id, g).. Setting p3 = Rm pa (id x çoN X id) and choosing suitable closed embeddings (pm :Al Rn we are led to the commutative diagram and yo N : N

M x N (ç°11/1

x id

(id, f)j

_

RV' x N

PZ

id x (id,

g)

RmxNxP 17,3

p Thus, we deduce the identity

(4.72)

g, o f,

p3 . o (id x (id, g)) o (witi x id) , o (id, f),

from (4.71) together with this diagram. Altogether, we come to the factorization (4.73) g* o f* = p* o(1)0. where 00 = ((pm, (19N 0 f, g o f) describes a closed embedding. From the other direction, we consider the expression

(g o f). p„ o

(4.74) using the closed embedding

q,i = ( çoM xid) o (id, go f) : M_, RmXRn x p .

180

CHAPTER 4.

MORSE HOMOLOGY THEORY

95t Relying on (4.65) we may once again set up a homotopy 00 cb i through 2 ( r' x ) , so that the assertion follows from closed embeddings M the homotopy lemma. Proposition 4.25 The Morse homology groups

Hrk`4"se(m) = Ern

f), k E No

can be organized as a family of covariant funct ors from the category of smooth manifolds without boundary together with smooth maps as morphisms into the category of Abelian groups. Moreover, homotopical maps induce identical group homornorphisms, i.e. Hk is a covariant functor for each k E No Ilk :

(C'-rnfds.,

homotopy classes of C' -maps

—■ AB

(m4N)

: Hk(M) Hk(N)) •

Proof. We have developed all the details except the explicit homotopy invariance for arbitrary smooth mappings. But it is due to the definition of b by means of the factorization that this follows immediately from the homotopy between the closed embeddings (id, f) (id, g) induced by f •--• g and from the homotopy lemma.

4.2.3

Relative Morse Homology

In this section we finally analyse the feature which is still lacking with respect to an axiomatic homology theory. That is the concept of relative homology groups associated to admissible pairs (M, A) of manifolds. The underlying idea is to deduce such homology groups from a chain complex, which is now generated by critical points of a Morse function f on M outside of A. The appropriate algebraic object is a quotient complex analogous to the singular theory,

C.K(f, fA) = C4)1 c . (fA ) . In this situation, the Morse complex C(fA) of a Morse function on A has to admit a canonical identification with a subcomplex of C,,,(f). Altogether, we once again require a functorial concept which guarantees at each stage compatibility with the axiomatically required long exact homology sequence. As to this concept of a relative homology theory, we first have to compile some technical preparations.

4.2.

THE EILENBERG-STEENROD AXIOMS

181

Relative Morse Functions and the Relative Morse Complex Definition 4.26 We call a pair of smooth manifolds without boundary (M, A), where A is a submanifold of M, admissible, if either A is already a closed submanifold or the topological boundary aA is a I-codimensional, orientable, closed submanifold of M. If (M, A) is admissible and of the latter type, we call a smooth function f E C"(M, R) steep w.r.t. aA, if —V f represents an inner normal field with respect to aA, that is: • V! rh

aAC

M and

• the negative gradient flow of f on

aA

flows into A.

Since we have already treated sufficiently the aspect of suitable Morse functions for closed submanifolds, we shall henceforth consider the situation of admissible pairs of manifolds where A is an open submanifold whenever we do not give an explicit specification. Remarks (a) The just defined steepness of a function with respect to aA is obviously a convex property, that is: If f,g E Coe (M, R) are steep w.r.t. OA, the same is true for all (1 t)f + tg, t E [0, 11. —

(b) If f is steep w.r.t. OA, it holds that Crit f n aA = O. Moreover, since M is a normal topological space, there is a neighbourhood U (A), which separates A from Crit f, U (A) n Crit f = O. Definition 4.27 We now call f E C"(M, R) a Morse function on (M, A), if f is a Morse function on M and additionally steep w.r.t. OA. In the situation of a closed submanifold A c M, instead of steepness, f has to yield a subcomplez C(f I A ) of C.( f) in the sense of Corollary 4.17, i.e. there are no f -trajectories leaving A. In principle this definition establishes the coercivity property for the Morse function f on M relative to A. Since due to the above assumption A is an 0 open submanifold of M, fi A cannot be a Morse function on A if OA holds. In this case the coercivity property is no longer satisfied, as not all sublevel-sets are compact. 10 However, since we are able to isolate the critical 10Referring to Definition 2.40 concerning the compactness analysis we restate that we have to consider manifolds with complete Riemannian metrics. Since the submanifold A with non-void boundary cannot be complete with respect to the induced Riemannian metric, we have to alter this metric near the boundary. Similar to the following argument, this can be accomplished without changing the Morse-homological information, i.e. Crit.(fA) and the connecting isolated trajectories.

182

MORSE HOMOLOGY THEORY

CHAPTER 4.

points of f from the boundary OA, it is possible to transform f 1A on a suitable neighbourhood of aA to a Morse function fA such that the respective Morse complex, that is, the critical points and the associated trajectory spaces within A, remains unchanged.

aA c Af

Given the closed submanifold

bourhood :

we choose a tubular neigh-

WCM

V

aA and a Riemmanian metric on the normal bundle v. Then starting from

ArIc3 = { vx E v

OA

< c(x) for all x E

8,41,

E

(aA, (o,i))

ça induces a topological basis of neighbourhoods { /Vc = ça(A11,3 )},<..." of aA for = 60(i9A) E C °° (aA, (0, 1)) small enough. Lemma 4.28 Given a Morse function f on (M, A), there is an c E

(0, 1)) ,

E

(aA,

co and a Morse function f A on A, such that fA i A \ Arc

=

f Av vc and Crit fA

n ivc — 0 •

Proof. Due to the assumption f is steep w.r.t. aA and {/Vc } c„‹...co represents a neighbourhood basis of a.A. Hence, referring to remark (b) above we may start from an co E C°c (aA, (0,1)) such that Crit f n Arc = 0 for all E Eo. Setting /V = N, n A, we define the smooth function g(,) : cp -1 (/V)

R, 1

vx

e IIvII

1

(x) -11vAlx

on ço -1 (N,±) = { vx E v+ I livrils < e}. Thus, composing g, = g,o o -1 , we obtain a function g, E Ccc (N,+ , (0,1)), such that

:A f, (x)=

R, f (x) + g,(x), x E N,± x E A\Ar, (x),

is well-defined. Due to the construction of g,, L is a smooth and coercive function on A, which is identical to f if restricted to A\N,. Since -V f represents an inner normal field of aA, we may choose E small enough, such that V if does not vanish within AT. ,± , too. Thus, fA -= h is a suitable Morse function on A according to the assertion.

4.2. THE EILENBERG-STEENROD AXIOMS

183

Definition 4.29 Let (M, A) be an admissible pair of manifolds. Then we call two Morse functions f and f' on (M, A) equivalent relative to A, if they are identical on M\A,

f

ff defst f M\A

ff l M\A •

The equivalence class [PA is called a relative Morse function on (M, A). Note that this definition remains reasonable in the case when A is a closed submangold of M. Lemma 4.30 We can identify the set of relative Morse functions {

[PA f a Morse function on (M, A)}

with the set of smooth functions f E C"(M\A, R) , 11 which - in the situation of an open submanifold A_- are steep w.r.t. 3,4, coercive on M\A and whose critical points within M\A are regular. Proof. The proof of this lemma is straightforward if one uses the extension results from the first section of this chapter. Example In the final chapter we shall briefly touch on the question of the representation of characteristic classes of vector bundles within the framework of this Morse homology theory. Let 7r : E M be a smooth, n-dimensional vector bundle. Then, as we already know, any Morse function f on M together with the quadratic form qn with respect to a Riemannian metric on E induces the Morse function f e qn on E together with the canonical identification

Crit. f

crit.(f (13 , qn) •

In contrast let us now consider the negative quadratic form -qq . It yields a fibrewise relative Morse function -qn,x on (Ex , Ex \U (Ox )) for each suitably small, connected and closed neighbourhood of zero U(0 x ) such that this pair is admissible. By analogy with above, we obtain a relative Morse function for the admissible pair (E, E\U (M)), where U(M) C E is a suitably regular but arbitrarily small closed neighbourhood of the zero section. Moreover, we once again obtain a canonical identification of critical sets, but this time with a shifting in the grading, Critf

Crit.±n(f e ( - qn))

We guess that this is the mitai step toward the Thom isomorphism. I li.e. smooth on a neighbourhood of M\A within M

CHAPTER 4. MORSE HOMOLOGY THEORY

184

Definition 4.31 Due to Lemma 4.28, we can find such a pair of Morse functions (f, f4 on (M, A) for each relative Morse function IfJA, that the Morse complex C.(f A) represents a subcomplex of C.(f). This means that we obtain a short exact sequence of chain complexes:

C(fA)

C(f)

C.(f)/C,,(f A )'° -

Consequently, we can define the quotient complex Ck ([f]A) = Ck(f) fA) =

Ck(f)Ic k (fA ), k E No ,

where Ck(f, f A) is generated by the critical points Critkf\CritkfA lying within M\A. Since these critical points are already uniquely determined by the relative Morse function [f]A, the relative Morse complex C.([f]A) is well-defined, i.e. independent of the concrete choice of the pair (f, f4 representing the fixed relative Morse function. We notice that due to the assumption of steepness w.r.t. aA, the 8-operator on C.(f, f A) is already uniquely determined by It is now this relative chain complex from which we gain the relative M \A. homology groups associated to MA,

k( f f A) = k (VIA) -= H k (C (f, fA)), k E No , together with a long exact homology sequence

Hk(fA) (‘

)

; Hk(f) (.1; Hk(f, fA)

14-1(f4

•••

-

We finish this introduction of a relative Morse complex with the following lemma: Lemma 4.32 There is a relative Morse function [f]A for each admissible pair of manifolds (M, A). Proof. In the situation of a closed submanifold A C M we may refer immediately to the extension result in Proposition 4.16. This is due to classical Morse theory [M1], which states that we can find a Morse function on any given smooth manifold. Let us now regard the case where A is an open submanifold We use the condition that the normal bundle of aA within M is orientable, so that we may construct a smooth function on the tubular neighbourhood of OA, which is coercive and steep with respect to A. Again due to the extension lemmata we can extend this function to a Morse function on (M, A). 0

The Relative Homology Functor The next step is to deduce independence from the concrete relative Morse functions as in the situation of the absolute Morse homology groups. Actually,

4.2.

THE EILENBERG-STEENROD AXIOMS

185

the possibility of change between suitably chosen Morse functions step by step allows us to build up a functorial concept. Now, the analogous limit process of the identification of the special homology groups depending on the functions becomes a bit more involved. Namely, we additionally require compatibility with the long exact homology sequence specified above. Definition 4.33 Let [PA and [g]A be relative Morse functions on (M, A). We call a homotopy [htiA satisfying

ht : M\A —R, t E [0, 1],

f

ht

'g I m\A = I m \A

a relative Morse homotopy w.r.t. A, if there are representatives f, g and ht on ht M such that ht describes a regular Morse hornotopy f ''.' g and ht is steep w.r.t. aA for all t E R. Lemma 4.34 Every two relative Morse functions [f ] it and [g],4 admit a relative Morse homotopy [ht]AProof. Considering the regularity condition for Morse homotopies from Definition 2.29, we begin construction with the trivial homotopy

4 = OW - f + (1

—

OM)

-

g .

Here, )3 : R —■ [0, 1 1 denotes a smooth function satisfying

(3(0 = { 0,

ti.

Since steepness is a convex property as we already remarked above, the requirement that 4 be steep w.r.t. aA for all t E R is satisfied. Thus, it particularly holds that Crit lq n u(aA) =0 for all t E R for an appropriate neighbourhood of 0A. Then, given a suitably small N, C U from the neighbourhood basis specified above, we define the following function: h: OR x N,) U (R\ (-1, 1) x M)

■

—

R

t 1 t ?.... 1 hci(x), x E N, . f(x),

—

h(t, x) ={ g(x),

This can finally be extended yielding a regular Morse homotopy such that the steepness condition is in fact fulfilled. CI

186

CHAPTER 4. MORSE HOMOLOGY THEORY

Lemma

ht

g of a relative Morse homotopy [ht]A admits Morse functions IA, gA and a Morse hornotopg hA,t on A such that the identity 4.35 Each representative f •--'

ha,ti Avv, = ht1 AVV,

holds for some f E C°° (024, (0,1)). Proof. The proof of this lemma may be carried out by the same means as in Lemmata 4.28 and 4.321. D Thus, the induced homotopy morphisms C(f) C(g) and ,thlt,t : c* (f A ) _,, C* (g A )

satisfy the consistency condition . ht IC ■•(f A) = 4)h" •

(4.75)

q)

Hence, we obtain a morphism between the respective short exact sequences of chain complexes, 0

• C(fa)

—

,.- Cs(f)

-

14AA,, a3,

1 4,ht

(4.76) 0 --.- C* (9A )

i. • C*([f]A) --• 0 e

14,[htiA

- C* (g) —,.- C*([glit) —.- 0

Due to the steepness condition of h t , 4>Iht 1 " is well-defined, independently of the actual extension (ht, hA,t). This morphism of short exact sequences of chain complexes gives rise to a homotopy morphism between the associated long exact homology sequences •

(4.77)

-6.-

Hk(fA) -

■

ke

1

40 t. A :

' ''' I Ik(g A) ' I I k(g) --i- Hic(g 7 g A) 'I I k - 1(gA) -1"

However, one problem remains. On the one hand, 4),,hA ' t and .71) /,:t fit in with the functorial concept, by means of which we establish the identification process. We consequently obtain independency from the concrete Morse functions,

187

4.2. THE EILENBERG-STEENROD AXIOMS

and we conclude from the algebraic five lemma that the (I)Lhti A describe isomorphisms, too. On the other hand, we cannot necessarily conclude the crucial functorial condition

4, 1h1/3 1 A

IA

0

[

h1 1 A

(4.78) luH,(f,fA) for these isomorphisms. This has to lead back once more to the deduction from the A-homotopies and the associated chain homotopy operators. Resorting to the methods of extension in the previous lemmata, we can show that, given any two relative Morse homotopies [hr]A and [h]A, there are pairs of representatives (ho, ho,A) and (h1, hi,A) as well as a pair of A-homotopies (HZ , HV) between these pairs of Morse homotopies, such that the condition of compatibility

C(f) sA ct

(4.79)

C.(fe() 14,0c, c.+1(f '3 )

is satisfied by the associated chain homotopy operators tlifia and Definition 4.36 As a consequence, relation (4.79) induces a chain homotopy operator also for the relative Morse complexes 11) S recit

:C.(r,g)

with the same properties as in the absolute case. Hence, the isomorphism (1.1,,h1 A of the homology groups is independent of the actual relative Morse homotopy, and relation (4.78) is true. Thus, the family

{H «(f f A) I [f]A rel. Morse p t.}

Dfr j A .

meets the identification requirement by means of the isomorphisms ( Moreover, this process is compatible with the respective long exact sequence,

Hk(M, A) = liM Hk(De 1A), k E No , so that we obtain the long exact sequence of Morse homology groups

...

H k(A)

H k(M) L Hk (M , A)

Hk_i(A)

...

188

CHAPTER 4.

MORSE HOMOLOGY THEORY

The only feature which still has to be verified is the identity of the homomorphism (i.). induced by the pairs (f, fA) and the homomorphism it , which was directly deduced above by means of the factorization techniques. This is not obvious, as we generally have to start from open submanifolds A c M with regard to the relative Morse homology.

Lemma 4.37 Both homoinorphisms (i„) , and i, deduced from the inclusion A

M are identical.

Proof In the case of a closed submanifold, the proof was carried out above in the auxiliary Proposition 4.22. Hence, let A be given as an open submanifold. Recalling the definition, we obtain

(4.80)

it = p„ o ((p x idm). o (idA x i). : H(A)

11,,(M)

where go represents a closed embedding ço : A c—+ Rk . Let (f, f) be a representing pair belonging to a relative Morse function on (M, A) and let g A = (cp x idA)„fA be a Morse function on Rk x A induced by the closed embedding (p x idA. We may construct a Morse function g on k x M in an appropriate way, which is steep w.r.t. Rk x aA and yields a relative Morse function together with g A on (R ic x M, Rk x A) in the sense of Lemma 4.28. Since A is embedded into Rk as a closed submanifold, we can likewise construct a relative Morse homotopy (ht] A represented by a pair (ht , hA,t) g

gk

e f, 9A

11.4)

qk

e fa

The first step of the proof now consists of the following commutative diagram

(.f A)

(4.81)

x idA).

({0} x id A) (DhA

H,(qk fA)

*

H., (gA) •

This follows from the homotopy lemma because {0} x idA and go x id A are homotopical closed embeddings of A into Rk x A. Fortunately, it does not matter that the associated homotopy of embeddings does not directly induce the previously fixed Morse homotopy hA, because the homomorphism 4)ht A on the level of the homology groups is independent of the concrete homotopy. The next diagram is an immediate consequence from the morphism (4.77) of the

4.2.

THE EILENBERG-STEENROD AXIOMS

189

long exact sequences: ,:khA

II(qk

(4.82)

((id

e f A) ._.____

x i),,) * 1

e

H.(94

I ((id x i),)

43 h.

11 .(9k

0 f)

. .

H(g) .

According to the definition, the homomorphisms (yo x i)„ from the closed embedding and ((id x i)„) * o (q) x idA),, are identical. Therefore, the identification process by means of the 4) h together with the combination of (4.81) and (4.82) imply the identity

(4.83) (cp x i) * = ((id x 0,

) *

o ({0} x id.) * : H* (A)

H* (R k x M) .

Since it holds that

(i.)4 = p* 0 ((id x i),,) * o (101 x MA) * : H4 (A) — 4 H* (M) due to the definition, the assertion follows immediately from the composition of (4.83) with the projection map p: Rk x M M. D In order to complete the deduction, we still have to go through the steps where we develop the relative homological functor by analogy with the absolute case. Moreover, each step must be taken in accordance with the long exact homology sequence. This means that we first have to consider closed embeddings 92 : ( M, A) -■ (N, B)

ço : M ‘- N closed

and cp(A) c B ,

and then projection maps p: (M xN,Mx B) — ■ (N, B). Finally, we must go through the factorizations

(M, A)

((id, f), (i, fl)

e

(M x N,M x B) -i The crucial point is the possibility of changing the relative Morse function via an appropriate homotopy, so that the associated isomorphism (1)!,h1A becomes

190

CHAPTER 4. MORSE HOMOLOGY THEORY

the identity after the identification process. However, by analogy with Lemma 4.37, we have to verify again that homomorphism (j.) * is identical to homomorphism j* , which is induced by the inclusion of pairs j : (M, O) (M, A).

Actually, all proofs within this framework for relative Morse functions can be carried out analogously by means of the extension and homotopy techniques. Summing up, we obtain Proposition 4.38 The relative homology groups (Hk(M, A)) for admiskEN sible pairs of manifolds and the long exact homology sequence fit in with the concept of a family of covariant functors Hk

:

admissible pairs of mfds., smooth mappings of pairs ((M, A)

(N, B))

AB,

k E

(f. : H k(M ,

Hk(N, B)) •

By analogy with the absolute case and relying on the methods of relative Morse homology, we are able to prove the homotopy invariance: Homotopical mappings of pairs f g: (M, A) (N, B) induce g. for the long exact homology sequences.

the same morphism

With regard to the intended axiomatic homology theory, we finally have to verify the excision axiom and the dimension axiom. The Excision Axiom Proposition 4.39 Let (M, A) be an admissible pair of manifolds and let B be

a closed subset B cia such that (M\B,A\B) again represents an admissible pair. Then the inclusion i: (M\B, A\B) (M, A) induces an isomorphism : H.(M\B , A\B)

H .(M , A) .

Proof. Obviously, it is sufficient to treat the case of an open submanifold A. According to the concept of relative Morse homology, we can choose a concrete, appropriate Morse function in order to compute the inclusion homomorphism and the actual homology groups. Considering a Morse function f E Cffor,e (M\A,R) such that the underlying relative Morse complexes of the respective homology groups are generated by the same set of critical points of f within M\A, we deduce the assertion immediately from the canonical

isomorphism C.(fm, f A) L' Cs(fm\s, fA\B) •

4.2.

191

THE EILENBERG-STEENROD AXIOMS

The Dimension Axiom

The precise dimension axiom in the framework of Morse homology is a trivial fact: Each Morse function on a space consisting of one point {x} has exactly one critical point, and this point has index O. Hence we obtain the identity H({x})

(Z, 0, ...) .

The argument for a finite-dimensional vector space, for instance I n, is similar. Let gn be the quadratic form with respect to a scalar product. Then the only critical point is a minimum and it generates the homology H ,,(Rn )

(Z , 0 , .) .

We shall now verify a relation which in fact is not trivial, but which is characteristic for any standard homology theory on manifolds. Proposition 4.40 Any smooth and connected manifold M satisfies tion

H °

the rela-

(M) m Z .

The main part of the proof, in which the property of connectedness plays a crucial role, is based upon the following M) Lemma 4.41 Let y,y' E Critof be local minima and let 'y E C° ({-1, homoa division y, -y(1) = y'. Then there is -y(-1) = path, be a connecting topical to -y consisting of isolated trajectories ui =

E

(xi , yi ) E Critif x Critof

This means that there are isolated trajectories uo, uo( — oo) = y, u(+00) = uo_1(—oo), i = 0,

, uk satisfying

, k —1 and uk(-1-00) =

such that the composition of suitably reparametrized curves ui is homotopical to ( 'di) • . . • fsk) = 7 • Proof. Since f is coercive, i.e. Ma is compact for all a E R, the number

v(7) = # { X E Critif I f(X)

f °

is well-defined and finite. The proof of this lemma is carried out by induction on Within the following steps of the proof we shall make use on the one hand of homotopies, which may be specified concretely in local coordinates, and

CHAPTER 4. MORSE HOMOLOGY THEORY

192

on the other hand of the global, negative gradient semi-flow on the compact subset Ma Alf : R x Ma Ma with a> maxfo-y

[—Li]

The beginning of the induction follows immediately from the application of a well-known lemma from classical variational calculus (see [Coup, from the so-called mountain-pass lemma: Given two different local minima y and y' of a coercive, smooth y, y' satisfying the function f, there is a further critical point x minimax-property

1(x) = inf max f (z), pEP zEp

P= {pCMI y,

E p, p compact and connected } .

In our situation we can additionally compute the Morse index, that is x E Critif. Thus, the beginning of the induction by v(ry) = 0 implies the identity of the endpoints y = y'. Regarding the induction conclusion, we may assume without loss of R takes its maximum in exactly generality that the function f o -y : [ - 1,1] and that one 'moment' to E 11 [ -1 ,

(4.84)

'y(to) E Critif

Otherwise we consider a homotopy of -y by means of the flow lIf T , so that max f ory decreases strictly until the maximal point is close enough to a critical point such that we are able to use the local coordinates provided by the wellknown classical Morse lemma (see [M1]). If the respective critical point should have a Morse index larger than 1, we find a homotopy in these coordinates, which 'pushes' -y past this critical point, so that the new curve 'y no longer intersect the stable manifold of this critical point. Note that this is due to the codimension of this stable manifold which is exactly the Morse index. This process can be iterated finitely many times, until condition (4.84) is fulfilled at last. If there are more than one to G [-1, 1] with f (-y(to)) = max f o -y, we will go through the following process sufficiently often. Given critical point x = 'y(t0) E Crit i f, we split 1, into parts 1, = The Morse index p(x) = 1 implies that there are exactly two isolated 'y. 'y. trajectories ur v , , uxy, of the type specified in the assertion, which leave x. Regarding the homotopy class, (4.85)

'y

'yyx Uxy ,

-1

-1

Uxyi Uzy2 • U sy2 • ^/xy ,

const = x is contractible. Analysing the situation at • IL ;y12 because x by means of the local coordinates provided by the Morse lemma, we observe

193

4.2. THE EILENBERG-STEENROD AXIOMS

that the order of the parts ux±y1. can be chosen in such a way that the negative gradient flow AP, pushes the curves = 'yy x uzy , and 'Y2 = u ryl,. ry.y, below the level 1(x), that is,

(4.86)

max f

-y1 ,2) <1(x)

I

for some

T

>0

.

After finitely many division steps of this type for each to, the parts -yt , which still do not necessarily consist of isolated trajectories, satisfy the condition

< v(-Y) upon which we have founded the induction process. Corollary 4.42 Given any two local minima y and y' for the Morse function f from the same connected component of M, there is a connecting curve consisting of isolated trajectories of the type E Jk/q

(Xi,

Yi) E Grit' f x Critof

Proof of proposition 4.40. This corollary yields the fact, that each two local minima of a fixed Morse function y and y' generate the same cyclic subgroup of Ho, {y }

= ± {V} •

It thus follows that

Ho(M)

(4.87)

Zp or Z .

This is concluded completely from the condition of connectedness on M. On the other hand, we have already concluded the fact that Ho(M) is infinitely cyclic from special Morse functions in Proposition 4.12. This led back to the problem of orientation. Resorting to the functorial concept, we are now able to prove this in general as a consequence of the dimension axiom. Let e: M —+ {x} be a constant mapping. We consequently obtain a non-vanishing group homomorphism

c. : Ho(M)

Ho({x}) ==-2 Z .

Hence, c. 0 together with result (4.87) from connectedness yields the asserH0(M) Z. tion

4.2.4

Summary

Summarizing the above results, we have analysed all elements of a homology theory which satisfies the axioms set up by Eilenberg and Steenrod. We have

194

CHAPTER 4. MORSE HOMOLOGY THEORY

obtained a family (Hk)kEN0 of covariant fun ctors from the category of admissible pairs of smooth manifolds (M, A) together with the smooth mappings of pairs : (M, A) —) (N, B), (p(A) C B as morphisms into the category of Abelian groups, so that the following axioms are satisfied: 1. The existence of a natural, long exact homology sequence, 2. the homotopy invariance, 3. the invariance under excision and 4. the normalization of the point space:

k=0 zo: sonst

Hi, {x}) (

4.3

The Uniqueness Result

In this section we will investigate the isomorphism result, which relates the Morse homology constructed in this chapter to any other axiomatic homology theory such as, for instance, singular theory. This equivalence between homology theories will be accomplished by means of the homology axioms verified above. Following Eilenberg and Steenrod [E-SI we will draw this final conclusion from the axiomatic approach in terms of a uniqueness result, which expresses the existence of a natural isomorphism to any other axiomatic homology theory provided that both theories in question be considered for appropriate categories of topological spaces. This so-called uniqueness category 12 in the case treated within this work will be the category of pairs of finite CW-complexes. We state the main conclusion as follows: Theorem 9 Let ]-1,. be any axiomatic homology theory defined on the category of finite CW-pairs. Then given a fixed isomorphism h0 : 4°' ({pt})

1-10{{pt}}

for a fixed point manifold M = {pt}, there exists a unique, natural equivalence of homology theories

eorse 2_4 ñ

with

(1)0 ({pt}) = h o ,

where 11°r" and 1L. are restricted to admissible pairs of manifolds which are finite CW-pairs. 12The reader is referred to (E-Si.

4.3.

THE UNIQUENESS RESULT

195

For one who is not familiar with this axiomatic approach set up by Eilenberg and Steenrod it might be surprising that no natural homomorphism defined explicitly in terms of the two homology theories in question is needed. This arises from a constructive existence result for just such a natural homomorphism which automatically implies uniqueness. The strategy for proving the theorem is to exploit this uniqueness property of the category of CW-spaces. But in order to make use of this feature we at least must extend the Morse homology deduced for admissible pairs of manifolds to a CW-subcategory which gives rise to the same uniqueness result. Therefore, we will restrict ourselves to the following pairs of CW-complexes (X, Y). Definition 4.43 Let us denote by CW „g the subcategory of CW-pairs which are embedded smoothly into finite-dimensional manifolds M, i.e. the characteristic maps associated to the cells are relative diffeomorphisms. Additionally, these CW-pairs must have open neighbourhoods (U, R) in M which are admissible and of which they are strong deformation retracts. Remarks • Without loss of generality we may assume that any two of such neighbourhood pairs are homotopically equivalent by smooth homotopies. • If we focus on manifolds M endowed with such a smooth CW-structure, M = {Xic }k=0,...,n we see that the pairs of skeletons (Xk, Xi) have this regularity property because of the well-known theorem that subcomplexes of CW-complexes are strong neighbourhood deformation retracts. Finally, this class of objects in CWreg will suffice for providing the proof of the main theorem, and this subcategory of sufficiently regular CW-pairs lends itself to the following extension of the Morse homology as an axiomatic homology

functor. Definition 4.44 Given a pair (X ,Y) c M from the category CW „g we define Ax,y) to be the family of neighbourhoods (U, R) of (X, A), of which (X, A) is a deformation retract in the above sense and which are itself admissible pairs of manifolds. Observing that il(x , y ) is a directed, quasi-ordered set by inclusions, which by assumption are homotopy equivalences, we define

(x, y)

lim

H.Morse (u it) •-•••

(U, R) c 1.1(x ,y)}

as the inverse limit with respect to the inclusion isomorphisms

i.(uv:sR? :

R)

Hs (V, S), (U, R) c (V, S) .

196

CHAPTER 4. MORSE HOMOLOGY THEORY

Remark There is some minor technical work left to show that the property of these neighbourhood pairs to be admissible does not disturb the directedness of

By a restriction to neighbourhoods which are homotopically equivalent to (X, Y) by inclusion, we are able to deduce the natural isomorphisms : ft* (X, Y)

H* (U, R), (U, R) E it(x,y)

immediately. This feature guarantees that exactness of sequences is preserved under this special inverse limit process. In general, this is not true and usually has to be provided in terms of a Mittag-Leffler condition. Obviously, by tubular neighbourhood constructions, we have a natural isomorphism fi* ( Al, A) mitorse ( Al, A) for pairs (M, A) which are equally admissible and regular CW-complexes due to the above definition. It is straightforward from the properties of the inverse system defined above to derive a functor ft* for the pairs of the category CWreg . Actually, by noting that the restriction to the homotopically equivalent neighbourhoods yields the exactness of the long exact homology sequence

one is able to verify easily the axioms for this extended homology theory ft.. In the future, we will drop the indices and Morse from the notation and use H* with a sense depending on the context.

We now move toward the construction of the natural homomorphism based upon the category CWre g • Proposition 4.45 Let H * and 114, be two axiomatic homology theories defined

on CWreg and let

ho : Ho ({pt})

111 ({Pt})

be an isomorphism for a fixed point manifold {pt}.

Then there is a unique

extension of h o to a natural isomorphism h* : H* Proof. The main step of the proof relies on the construction of CW-homology. As this is standard knowledge from algebraic topology the reader is referred to any fundamental book on this topic. At first, we will deduce unique isomorphisms h* for n-cells en and n-spheres Sn from the given homomorphism ho . This will be done by means of the homology axioms and particularly by induction based on the MayerVietoris sequence, for which exactness holds because all pairs appearing in the

4.3. THE UNIQUENESS RESULT

197

following argument give rise to proper triads. Let us recall this essential and constructive tool from homology theory. Given a proper triad (X; X1, X2), i.e. the inclusions

: (x1, X1 n X2) "4 (X1 U X2, X2) i2 (X2, X1

and

n X2) "4 (X1 U X2, Xi)

inducing isomorphisms of the homology groups in all dimensions, together with relations X = X 1 U X2, A = X1 n X2, the associated Mayer-Vietoris sequence 13

Hq (A)

Hq (X i ) Hq (X2)

Hq (X)

Hq_ 1 (A)

is exact. Now, by naturality with respect to the homology axioms, it is straightforward to construct isomorphisms

h. : H(X)

H:(X)

for all spaces X which are finite disjoint unions of open or closed n-cells. Let {x} be the unique X be a single point space X = {x} and let ço : {1)0 diffeomorphism. This gives rise to

ho({x}) = (p.o ho o

: Ho ({x}) !=-% H({x})

in a unique and natural way. From the Mayer-Vietoris sequence we deduce h. for all finite disjoint unions of points and the result for n-cells follows from the homotopy axiom. The next step is to construct the extension of these h. on arbitrary cells to arbitrary n-spheres. This can be done inductively by means of the Mayer-Vietoris sequence with respect to the standard decomposition of the n-sphere into the two closed hemispheres D+ and D - with the equator sn-1 = D+ n D- as intersection. Again by naturality and induction we find unique isomorphisms h. : H,.(X) =.■ H. (X), where X is allowed to be an arbitrary n-cell or n-sphere from CW„g . Further, making use of the axiomatic long exact homology sequence we infer the extension of h. to all pairs of the diffeomorphism type of (Du, Sn -1 ) denoting the closed n-disk with its boundary sphere. Finally, the Mayer-Vietoris sequence again implies the extension to one-point unions of n-spheres V = 1 S2 , which are objects in CW„g , also. We now have developed the natural homomorphism h. for a family of cellular spaces already large enough to carry out the crucial step toward CW-homology. We now denote by X a cellular complex from CW„g . As is well-known from standard algebraic topology, the groups

C(X) = Hq (X", Xq -1 ), q 0 13For the definition of XIi 1) and A and more details, see [E-S]. ,

198

CHAPTER 4. MORSE HOMOLOGY THEORY

are free Abelian of finite rank, where (X 9 ) q)0 denotes the family of q-skeletons of X for q O. If Fe: (Dq,Sq -l ) (Xq, Xq -l ) describes the characteristic map associated to the q-cell e C X, we can take the family { F:({Dq} sq -i) E Hq (Xq, X q-1 ) e q-cell in X } as a basis for Cq (X). By means of this basis, the homomorphisms F: and the above investigations about the extension of h, to finite unions of pairs (De, 5q -1 ) and one-point unions of spheres, we immediately deduce natural isomorphisms

Hq'(Xq,Xq -l ), q 0 .

hq : Hq (Xq,X")

Thus we have already established a natural equivalence between the CWhomology theories

(4.88)

Hq (C' (X)), q) 0

hq : Hq (C(X))

associated to H. and 11,,' . Moreover, we see from the above constructions that this equivalence is unique with respect to (ho , {pt}). Now, using the canonical and natural isomorphisms associated to CW-homology

0 and

T: Hq (C(X))

Hq (X), q

T' : Hq (C'(X))

H,' (X), q 0 ,

we can accomplish the proof by deriving the natural isomorphism h. between

and 11 on CWreg as the composition

(4.89)

h. --=

o

0 T -1 .

Here, kew means the cellular equivalence given in (4.88). The naturality of h* in general follows from the naturality with respect to cellular maps and cellular approximation together with homotopy invariance. Thus we have proven that CWreg represents a uniqueness category in the sense of Eilenberg and Steenrod. Let us now consider the above extension of Morse homology to an axiomatic homology theory on CW,-eg together with the uniqueness property of this category. Since this uniqueness category contains the admissible pairs of manifolds equipped with a finite CW-structure, we obtain the proof of Theorem 9. Of course, it should be possible to derive this uniqueness of Morse homology for a far larger class of spaces. But within the intended frame of this work we shall be content with this result.

Chapter 5

Extensions In this last chapter we present a brief outline of further elements of algebraic topology within the framework of Morse homology. In this context we emphasize the Poincaré duality arising rather immediately and the feature of product operations appearing naturally in the Morse theoretic approach. At first, we briefly introduce the dual concept of Morse homology.

5.1

Morse Cohomology

By strict analogy to other known homology theories which are deduced from chain complexes, we obtain a cohomological version of Morse homology by means of a dualization concept. Definition 5.1 Given a chain complex (Ck(f), ak)kEN. as considered in the context of Morse homology, we define a cochain complex (Ck(f), bk)ko as follows: cc(f) = HOM(Ck(f),Z),

( t5k_k ,

Xk+1)

(ak aki-lXk+1

for all ak E Ck (f), xk+i E Critk+lf, k E No • This definition is unique, as Critk+1 f comprises exactly the generators of

ck+i(f). Remark Discussing this dualization process we must be aware of the fact that Ck (f) need not necessarily be finitely generated any more, as we assumed

CHAPTER 5. EXTENSIONS

200

for Ck(f). Nevertheless, if the set Critkf is finite, we are supplied with the canonical isomorphism ck(f)

c" (f)

(5.1)

Ii, Xk

((xkl •)

Yk

Yk = Xk

otherwise

Moreover, we compute the coboundary operator exk

= E ( ay, xk

if in addition Critk +1 f is finite. In principle, we resort to the same isolated trajectories together with their characteristic signs determined by a coherent orientation as in the definition of the boundary operator. As indicated by the increasing Morse index, this amounts to replacing negative gradient flow by positive gradient flow. But two problematic items still remain. First, —f need not be coercive. We recall that it was coercivity by which we established the compactness result for the trajectory spaces. In fact, if the underlying manifold is not compact, changing from f to —f necessarily abolishes coercivity. The other problem arises in the case of non-orientable manifolds as remarked in Section 4.1.4. Actually, both flaws within our idea of symmetry concern exactly the requirements of the Poincaré duality. This will be analysed further in the following section. Summing up the present discussion we observe that the dualization concept known as the Hom-functor allows us to construct an axiomatic Morse cohomology. Actually, the application of this functor together with the known universal coefficient theorem gives rise to a transfer of the identification process (yielding cohomology groups independent of concrete Morse functions) as well as of all functorial and axiomatic properties to the contravariant functor H. The reader not familiar with this dualization process may refer to the analogous treatment of singular or simplicial cohomology.

5.2 Poincaré Duality As we noted above this Morse cohomology theory has the disadvantage that the cohomology classes cannot necessarily be represented by the critical points of a Morse function. This is merely possible when the cochain complexes are finitely generated. We observed that the compactness assumption for the manifold M within classical Poincaré duality provides a sufficient condition for this finiteness of the set of generators. Then, coercivity is trivial. On compact manifolds we are naturally provided with a symmetry which is expressed by the lack of a

5.2. POINCARÉ DUALITY

201

canonical distinction between negative and positive gradient flow. If we additionally assume the orientability of M, both requirements for Poincaré duality are complete. On the other side, the problem of the noncanonical construction method of a coherent orientation vanishes, with respect to the condition of admissible pairs of trivializations in Definition 3.7. Under these circumstances, Morse homology cannot be set apart from Morse cohomology with regard to the construction. We are provided with a natural isomorphism in the case of an ndimensional, closed and orientable manifold M

PD: Hk(M), Hn-1 c kl V

k =0,..., n

as follows: Let f be any Morse function on M. Then, —f is also a Morse function and we can easily find a regular Morse homotopy

f 4 -f This induces the canonical isomorphism

(5.2)

41) sh : Hk(f)

Hk(—f), k = 0, . , n

Noting now that we may use finitely many critical points from Critkf as generators of Ck (f)=Hom(Ck(f), z), we are led to the identity

(5.3)

Critk( — f

)

—

E

Critn—kf,

This identity gives rise to the likewise canonical isomorphism

: Ck(—f) 2=%

en—k W ,

a-

and 6-operator, equally. This means that it which is compatible with the is well-defined on the level of homology and cohomology. Thus we canonically obtain the isomorphisms

4). : Hk(f) r. :

Hk( — f)

Hk(—f) and Hn_k (f)

and hence by composition

(5.4)

PD =r. 0 4). : Hk(f)

+ H"(f), k = 0,

,n.

202

CHAPTER 5. EXTENSIONS

Actually, due to the construction, this proves compatible with the isomorphisms induced by Morse homotopies leading to the identi fi cation processes H(M) = lim H„(M , f), etc.

Altogether we naturally obtain the Poincaré duality for closed and orientable manifolds. The same is true for the case of Z2 coefficients where non-orientable closed manifolds are admitted. Then, the problem of a coherent orientation becomes insignificant.

5.3

Products

Here we recall the naturally induced construction of the Morse function f e g on the product manifold M x N, provided that we are given the manifolds with Morse functions (M, f) and (N, g). In the last chapter, we have already benefited from the canonical identification of the critical points of this induced Morse function with respective pairs of critical points of single Morse functions. If we now use the tensor product for chain complexes in the form (5.5)

C,(g)) k =

(C,.(f)

® C(g)

together with the boundary operator (5.6)

ak(0-i ®

Tk_i)

for

= ôf

0 0 Tk-i ( -1 )

i0.

i

aige_iTk - i,

cri 0 Tkj E Ci(f) Ck-i(g)

we are able to state Lemma 5.2 Let f and g be any Morse functions on manifolds M and N, respectively. Then, with regard to the induced Morse function f e g on M x N, there is an isomorphism between the respective chain complexes for any arbitrary coherent orientations, C.( f ) C„(g)

C4.(f g) .

Moreover, this isomorphism appears to be canonical up to transformations of the coherent orientations. Proof As mentioned above we start with the identification

Critk(f ED g) = U Criti f x Crit g . i-i-i=k

5.3. PRODUCTS

203

Thus we define a chain group isomorphism : (C.(f) Ø C,,(g)) k

Ck(f e g), k

E

No

by means of the tensorial extension of the mapping xs

a(x,, yi ) (x$, Y.2)

defined on the generating elements (x,, y, ) E Crit, f x Crit j g. Here we denote by E {±1}C1it.(f@g) an equivalence transformation of the coherent orientation on M x N.

We now have to verify that 1-"• in fact yields a chain homomorphism, provided that we have chosen a suitable transformation a. This choice of transformation is related to the coherent orientations on M, N and M x N which are supplied in an noncanonical manner. As to the a-operator on C.(f g), let us consider the gluing situation sketched in Figure 5.1. Note that we

(Y.f,Yg)

Figure 5.1: Cobordism

equivalence in the product case

deal with isolated trajectories in M x N, that is 11 (Y...) = 1 • Due (5.7)

to our analysis of the gluing process we derive the equation TŒ (U f X g) •

ra (y.f, ug )

—Ta (X f Ug) •

Ta (11 f

yg )

in a way which is entirely independent of the coherent orientation on M x N. This means that (5.7) holds for all transformations a. The actual problem

204

CHAPTER 5. EXTENSIONS

consists of relating the characteristic signs To (Uf , x9 ), etc., of the isolated trajectories on M x N to the signs r(v.f), etc., of the associated trajectories on M and respectively N, that is, To (I f 19) = (17 (t t f X g)T (IL f

(p(x f

rc,(X f

Ug)

with

cp (. , .) E {±1}

and

?LOT(%)

Since we are free choose a suitable transformation a E {±1}c(M 9) , the relating signs (p(.,.) are independent of the concrete trajectories and critical points. They can depend merely on the Morse indices and - this is the crucial point - on the order of the pair of the isolated non-trivial trajectory and the constant trajectory. Actually, already in the definition of the tensorial chain complex C,,,(f) C * (g) we have different equivalent possibilities for the boundary operator. We have selected the option 0(x 1 0 X9) =- 49X f X9 + (

- 1) 44X f ) X f aXg .

Thus, choosing an appropriate transformation a, we obtain the relations Teti, f Xg) = r(Uf)

(5.8)

and

r(x1,u9 ) = (-1) 0(rf ) r(ug ) . Hence, the identity

xf

X9 = (X f

X9 )

holds up to equivalence transformations of the noncanonical coherent orientations.

Definition 5.3 Since Morse homology groups have been constructed in such

a way that they are invariant under equivalence transformations of the coherent orientation, the above isomorphism P. induces an isomorphism P* at the homology level. This isomorphism, which is therefore canonical, appears to be the cardinal feature, upon which we found the definition of the homology cross product. It is accomplished by means of the natural homomorphism A : Hp ( f) ®Hq (g)

Hp+q (C *(

f) ® C(g))

provided for any tensorial chain complexes. A usually appears in relation with the well-known Kiinneth formula. We define the homology cross product as

x = P*

: Hp (f) H q (9)

Hp+ q (f ED g)

•

Due to the naturality of the hornomorphism A we are able to transfer the entire identification concept based on the isomorphisms (13h* induced by Morse homotopies. Hence, we obtain the cross product for Morse homology independent of

205

5.3. PRODUCTS

the Morse functions: Hp (M) 0 Hq (N) 2, H p+q(M x N) . Since P. is a chain isomorphism, resorting to the duality functor Hom allows us to construct likewise a cross product for the cohomology groups: ( P-1) -

X : HP (f) 0 WI (g) -4 HP±q (C.( f)g C .(g)) --, H' (f

Once again we obtain the associated product for the cohomology independent of the concrete Morse functions, HP(M) 0 Hq(N)—', ' HP+q(M x N) . Definition 5.4 Having defined the cohomology cross product by means of a relation which is naturally immanent in Morse theory, we now are able to define the cup product. Here, we use a classical method based on the fact that cohomology is a contravariant functor. Namely, we consider the diagonal embedding

which is a closed embedding. Hence, it immediately gives rise to the homomorphism A* : H k (M x M) --* H k (M) . To give more details, we obtain this homomorphism from a Morse homotopy h, where A. f denotes the extension of the Morse function f EDq n from the tubular neighbourhood of the diagonal M to M x M. For {a} E Hk (M x M), A*{a} = IA'al E Hk(M) , where A•cr E Hom(Ck( Az) is determined by A*a, x k ) =( a,

et •

A(Ik))

for

xk

E Critkf -

Thus, we define the cup product as the composition HP+q(M x M) .1■ HP±q(M), HP(M) 0 H(M)—`-) '

Finally, we intend to present a brief outlook into the possibilities of extending this homology and cohomology theory gained from classical Morse theory even further, for instance toward the topic of vector bundles.

206

CHAPTER 5. EXTENSIONS

Proposition 5.5 Let M be a smooth manifold and let

it : E M be a smooth n-dimensional vector bundle on M. Furthermore, let E0 = E\U(M) be the complement of a closed neighbourhood of the zero section M within E, so that it is contractible to the submanifold M and (E, E0 ) is an admissible pair of manifolds. Then there is a natural isomorphism

T: Hi(M)

Hi+n (E Eci)

for all i G No.

Proof. Let f be an arbitrary Morse function on M and let qn be the quadratic form associated to a Riemannian metric on E. Referring to the example which we have already analysed with respect to relative Morse functions, we observe that f (—g n ) represents a relative Morse function on (E, E0) with its critical points identical to those within Crit. f lying on the submanifold M. In the same way we conclude that all isolated trajectories counting for the relative boundary operator of the relative Morse complex associated to f e ( — qn ) on (E, E0) are located on this submanifold. Actually, we immediately obtain a chain isomorphism between the absolute Morse complex of f on M and this relative complex. Since all stages of the identification process via Morse homotopies can be accomplished in a manner that is compatible with this construction, the assertion follows from this identification isomorphism T between the absolute complex of M and the relative complex of (E, E0 ). Here, the grading of the relative complex is shifted by the constant n given as the dimension of the vector bundle. Referring to the classical framework of characteristic classes one would now have to verify that this isomorphism T describes the Thom isomorphism. Moreover, the following diagram should be given:

Hi +n (E E0)

111 (M) Here, u E Hn(E,E0) denotes the Thom class of the bundle E. Let us finish the discussion with the guess that one should be able to obtain a representative for this Thom class within the framework of Morse cohomology theory by means of the critical points (xo, 0) G Critn(f. S ( - 472)) for xo E Critof.

Appendix A

Curve Spaces and Banach Bundles In this chapter we present a brief introduction of Banach manifolds and associated Banach bundles which form the foundation of the entire analytic approach to Morse homology chosen in this monograph. We discuss explicitly the structure of an infinite-dimensional Banach manifold defined on curve spaces as was stated in Proposition 2.7. The first section of this chapter deals with these manifolds themselves. The second section comprises an analysis of certain Banach bundles on these manifolds. We shall partly use notations and schemes of proofs taken from [Eli]. However, it is quite important to point out that our more specialized situation differs from that in [Eli] with respect to the noncompactness of the curves' domain R. There are several steps in the discussion where we need a more refined treatment.

A.1

The Manifold of Maps 'Px1: (R, M)

Definition A.1 A covariant functor

6 : Vecc, (R) — ■

Ban

is called a section functor if it associates to each smooth vector bundle e on a vector space 6(e) of sections in e together with a Banach space topology, so that 6 maps each smooth bundle hoinomoiphism : 77 to a linear map 6.60 E £(8(0; 8(71)) defined by (6.(p) • s • s, and C°3 (Hom(, TO) lendowed with the already defined differentiable structure

208

BANACH BUNDLES

APPENDIX A. CURVE SPACES AND

is continuous. Auxiliary Proposition A.2 The mappings 14,1,2,4 veccoo (-111)

Ban

defined in Definition 2.5 are section functors. Proof. We have already verified that the Banach space topology on 42 (e) and 0. Each pair of

Law is independent of the respectively chosen triviafization

norms 1H11,2 and 11'11L induced by such trivializations from the canonically normed vector spaces H 1 '2 (R,R 71 ) and L2 (R, RV), respectively, are equivalent, because 0 is equipped with asymptotical differentiability. Hence, we may start without loss of generality from the trivial bundle e = R x Rn, i.e.

H Fit '2(e) _

Rns)

La (e) = L 2 (R, Din) .

and

Now let A E C"(HOm(e, r))), that is, without loss of generality, A E C"(117k, M(m x n, R)). Then the following estimates hold for s E H 1 ' 2 and respectively L2 , where we denote (As)(t) = A(t) • s(t): (A.1)

1 AS 110,2 11(AS)' 11,2

=

11A1103 • 1 S 110,2 + AS/ 11123,2

= fR (1 111 81 2

IAS/ 1 2 + 2( A' s, As' )) dt

( IIA's11123,2 11A8/ 11123,2 2 ( IIA / 11123,2 Ils11 203 + 1011 .2.. 11s/11,23,2 2

(A.2)

)

•

According to Lemma 2.2, R-differentiability yields the finite norms 11111110,21 11A11. < Do. Moreover, the estimate (A.3)

11s11.

11s111,2

for all s E H1 ' 2 (R,Rn) .

follows from the simple calculation 1s(ti)1 2 — 1'9 401 2 ti

= ft.

fti

=

110

d

s(t)

2

dr

rt

J

(1 S(T)1 2 + 1 41 2 ) 114 21,2 • to

Thus (Al) and (A.2) give rise to the estimate 11 }1s11?.,2

const (11 ,4/ 11O,2 + 11/1 11 2.) 11s2

A.1.

THE MANIFOLD OF MAPS '1).y (lEt, M)

209

that is

(A.4)

il 11 ilL(H1 , 2;x1 ,2)

const

iiAil 200 •

Consequently, the mapping 6. : C"" (Hom(, 17)) uous.

L( 6 (); e(7/)) is contin-

Remark It is worth mentioning that the estimate (A.4) with 10'11 0,2 < oc relies essentially on the special choice of the differentiable structure on lik! Corollary A.3 As to the section functor 6 = Hill' 2 , the estimate IjAll„ for A E H 1,2 ( ,M(m x n,118)) implies the continuity of the map

6:

(110711 (e, 71))

which is stronger than the mere section functor property. The next step provides us with properties of this special section functor, which correspond to the conditions of a so-called manifold model in [Eli]. Auxiliary Proposition A.4 The section functor 6 has the properties

(a) 6() (b)

= H, llt 2 on Veccoo(R)

C°(e) is continuous for each e E Vecc— (R) and

e(Hom(e, 71))

is continuous

A (A. : s for all e,71 E Vecc—(R) .

As)

(c) Given e E Veccoo( -111) and an open subset 0 c e such that there is a section -y E CCV with compact support in R and -y(111) c 0, each smooth bundle map f E C(0,7) satisfying f(0+„) = 0+ 00 induces a well-defined and continuous map on 6(0) = { s E 6()18(R) c C.9 } 2 6 ( 0)

s

e(n) f s .

Proof. (a) has been already proved by (A.3) and (b) follows likewise from (A.3) together with (A.4). As to (c): Since f is a smooth bundle map, its fiber restrictions ft are in particular Lipschitz continuous, uniformly in t E R. Thus, with respect to any trivialization of e, we obtain the estimates

(A.5) 2Note

1ft(xi) — ft(Yi)1

const

that this subset of sections is open within

— yt1

for all t E R

6()

due to item (a).

210

APPENDIX A.

CURVE SPACES AND BANACH BUNDLES

and

fit (s(0) I

const I s(t)I + I ft (0)1 .

Tk,

Since ft(0) E 97), this estimate together with Corollary 2.4 implies that H 2 (q) is well-defined. The continuity of L. with the map L. : 14' (0) respect to the section functor 6 = is obtained from (A.5) together with an analogous uniform estimate involving first derivatives of the Ti-smooth bundle map f. El

42

Now these properties enable us to deduce the crucial lemma concerning the construction of the manifold of maps within our framework. Fundamental Lemma A.5 Let 6,0 C e and f E C(0, q) be as in the auxiliary Proposition A.4. Then the map f : 6(0) 6(n) is smooth and the k-th derivative is given by Dkf,,(s) = 6.(Fkf o s), which is well-defined. Here, Fkf : 0 —* Hom ( ED 77) denotes the k-th fibre derivative of f. ..

.

;

Proof. We prove the lemma by induction on k.

k = 0: The continuity of L. has been already verified as item (c) in the auxiliary Proposition A.4. k = 1: Let 0 be fibrewise convex without loss of generality and let so E 6(0) be fixed. Then, for x, y E 0 from the same fibre, i.e. 7r(x) = 7r(y), we define 8: 0 (A.6)

e(x, y) • z

—■

Hom(e, 71) [1. F f (x t(y — x))dt — F f (x)] • z

Since Ff : O —■ Hom, q) is smooth and fibre respecting, the same is true for 8. Moreover, we easily verify the equations

(A.7)

e(x, y) • (y — x) f(y) — f (x) — F f (x) • (y — x),

0(0,0) =0 .

The section functor 6 with properties (b) and (c) from the auxiliary Proposition A.4 gives rise to the composition of the continuous maps

6(8) : 6(0) ED 6(0) —■ 6 (Hom(, n)) —,.c(e(0 ; 6(n))

,

so that

(A.8)

6 ( 0) (80,80) = 0

and

lim 8-.30

e(e) (so, s) = o .

Since (A.7) implies the identity

— .f.(so) — 6.(F f 0 so) • (8 — so) = e(e) (so, s) (s

— so) ,

A.1.

THE MANIFOLD OF MAPS P'x' y (R, M)

211

the map ft is differentiable at so with the representation of the differential

(A.9)

Df.(so) = e.(Ff ()so) E .C(6(6); 6 (71)) •

It should be mentioned that, due to (A.4), this is also well-defined if it holds 6 (Hom(, ri)). The continuity of Df. : that Ff(±oo, 0) 0, i.e. F f oso 6(0) — ■ .C(6(6); '5(j)) follows from the continuity of the mapping

8(0) —4 6 (Hom(6, n )) c C(6(6); 6(n)) s 1— ■

Ffos—Ffoso

because

0 :0 — ■

Hom(, i) F f (t, x) — F f (t, so(t))

(t, x) 1— ■

is a smooth13 fibre respecting map, and it satisfies the condition 19(±oo, 0) = (±oo, 0) as with (c) in the auxiliary Proposition A.4.

k k + 1: Let us now start from the k-times continuously differentiable map together with f. Dkf,,(so) = e.(Fkf

so ) E £(6( ) ,...,6( ) ;e(n))

and let us denote by f (k) the map

f (k) f (k) (t,) = Fkf(t,) — Fkf (t, 80(0) Then f (k) again satisfies the inital condition on f, namely in particular

f (k) (±oo, 0) = 0 With 77„, = Hom(e ... ED ,71). Thus, the step 1 = l' yields the continuous L(6 (6), . . , 6(e); 6(77)) together with differentiability of f (k). : 6(0) Df(k).(.-9o ) =

(Ff(k) 0 go ) = 6,, (Fk±lf 0 go)

We now assume (M,g) to be a para.compact, Riemannian Cc°manifold together with the associated exponential map. Furthermore, let D denote an open zero section neighbourhood within the tangent bundle r : TM —■ M such that

(A.10)

D

6

V(A) cMxM (7(6),exP(e))

3Here, without loss of generality, so E Cw

(e).

212

APPENDIX A. CURVE SPACES AND BANACH BUNDLES

represents a diffeomorphism onto a diagonal neighbourhood within M x M. We observe that smooth, compact curves h E Coe(R, M) give rise to the pull-back bundles

h* T M = { (t, e) ER x TMIT (e) = h(t)} E Vecc— (R) ,

HIV

on which the above section functor is well-defined. From now on we shall again use the notation 6 for this section functor. We obviously obtain from D the open zero section neighbourhoods

h*D = {(t,e) EIR >
exp h s = exp o s E C° (, M), (expo s)(t) = exph(t) s(t) . Thus, provided any fixed endpoints x, y E M, the set of curves

I

Pzi :y2 (R, M) = 1 expo s E C° 0R, M) s E e(h*D), h E CZ(R, M) is well-defined. Theorem 10 Ps1,y2 (R,M) is equipped with the differentiable structure of an infinite - dimensional Banach manifold. The family

{ e(h*D), 6(exph )1 hECry (IR,M) where

e(exPh)

6(h *D) s

ex p o s

represents an associated atlas of charts. Moreover, the Banach manifold P1,112 together with this differentiable structure is independent of the Riemannian metric g on M. Proof. We consider the following smooth mappings with respect to h E Ca Oft, M): Çbh:

h*V

Rx M

(t,

(t, exPh(t) e) •

(A.11)

According to the choice of D, this is an embedding onto an open neighbourhood of the graph of h. We consequently define

= Oh(h*D) c RxM , M) graph g c Uh , OtT l o (id, g) E 6 (h*D) 64) = g E Lih

C C(, M)

A.1.

213

THE MANIFOLD OF MAPS /1) ,2y (R, M)

and

4 6 ( h 1 ) 6(1 4 ) —> e(h*D)

(A.12)

g

owl° (id,g)

It holds that 1 1:121 (1R, M)

U 6(14) hEcrv (re,m)

that { 6 ( 017 1 )} hEcwii,m) is a family of one-to-one correspondences. Let us now endow this curve space with a topology such that these bijections become homeomorphisms. The fact that this topology is well-defined is particularly due to the following proof of the diffeomorphism property for changes of -1 the charts 6(0-1 ) and

choose any two supporting curves f, h E CT.y (rk, M) with ( n1,1 f) C ._*_ h 'D and 0 f = (uh nuf) c f *D Oh (1)h— 1 1,Ih are open subsets satisfying the condition in (c) in the auxiliary Proposition A.4. The non-void intersection e(Lih) n e(tif ) is open within e(Uh) and 6(111), respectively. We now consider the mapping Let us

UhflUf ^ O. T _en

(1) .fh = f l 0

(A.13)

4 fh ( t 7

—

: h*TM

D

Oh Of C f *TM

t f exPf(10( exPh(t) .

)

which describes the change of the charts It fulfills the condition (I)fh(±co, 0) = 0 as we can see from the fixed endpoints h(±oo) = f (±oo) = x, y. Thus, the fundamental Lemma A.5 states that (1) fh* = 6(49 -f1 ) 0 6(0h-1 )

1 : 6(0h) —■ 6(0f)

is a smooth map with (Dfh47 1 = chf as its inverse, hence the diffeomorphism property.

compatibility of two atlases belonging to different Riemannian M) lies dense within Cx° ,y (E, M) metrics follows from the fact that c expg represents a local exponential g, the any arbitrary metric and that, for sense of (A.10). diffeomorphism in the Supplement Moreover, we are able to find a countable subset of Csœny (Irt, M) which lies likewise dense within CZ y (rt, M). This enables us to reduce the atlas given in Theorem 10 to a countable subatlas. The

Corollary A.7 Given Riemannian COE'Lmanifolds M and N together with a map f E CŒ) (M , N),

(f) . 771 :1,2 (R, m )

(y) (R, N)

p f0

^y

214

APPENDIX A. CURVE SPACES AND BANACH BUNDLES

is a well-defined, smooth map between Banach manifolds.

cy,s,,f(y)

Proof. First, any fixed -y E ny2 is mapped to f o -y E N). Since Crw,f(y) (II,N) lies dense within C7 (z),f(y) (ii,N), we find a representation of the shape

f o -y = expk t, k E C74 ),f(y) (R, N), t E 0(k*DN) . Secondly, we may express -y as

= exph s, h E

(R, M), s

E

e(h*Dm) .

Furthermore, we find an open neighbourhood 6h C h* DM, such that the conditions in (c) in the auxiliary Proposition A.4 are satisfied for the smooth fibre respecting map 11)

h = expk-l of o exp h : 6h k * DN

7

= ( 100 , 0) • Hence it is due to the fundamental Lemma A.5, that qh* is a smooth map. This proves the corollary.

A.2 Banach Bundles on Px1,y2 (

M

:

6(6h) —> 6(k*DN)

)

We henceforth use the notation 6 for the section functor . Let us fix a Riemannian metric g on the manifold M, as was required in the last section in order to define the atlas on Px1,y2 . Given such a g we are provided canonically with the following features known from Riemannian geometry:

Let K :T(TM) —> TM, T : TM —> M denote the unique Levi-Civita connection and the canonical projection in the tangent bundle of M. As a consequence, K and DT : T(TM) — ■ TM yield a decomposition of T(TM) into a horizontal bundle and a vertical bundle: 4

(A.14)

To(TM) = Tch (TM)

ker (137-( )

) CETM

= ker (K( ) )

4Note that the vertical bundle comes canonically and independent of any specification of a Riemannian metric.

A.2, BANACH BUNDLES ON P jey l (R,M)

215

Referring to the exponential map associated to g and K, exp : M, we obtain isomorphisms at any E D, where D once again denotes the injectivity neighbourhood associated to exp:

(A.15)

Vi exp(C) = DexP(C) ° (DTI Tc h(Tm)) 1 T,-()M ■ -1 172 exp(C) DexP(C) ° (KI T(Tm)) : Tr (c) M

Texp(C) M

Texp (c )M .

It holds that VI exP(0) = \7 2 exP(0) = idT, (o) m. Furthermore,

(A.16)

ê: D

Hom(TM, TM)

e(C) = (V2 exP(C)) -10 exP(C)

TM

Tr()M

is a smooth fibre respecting map which satisfies the identities

• 0(0) = id, • F0(0) = 0 and • e(C) . = C • Given any y E TM und X E C°" (TM), we obtain the covariant derivative of X in the direction of y from the formula

VvX(p) = K o D pX • v Thus, straightforward computation for h E CZ and the identity

E C'(h*TM) yields

— exp h = exP(C) • h + v2 exP(C) • VtC • at Let us consider once again the map from (A.13) representing the change of charts (I) f h for smooth rt-curves f,h E CTy M). Then the fibre derivative E Horn (Th(t)M, f (t) Al) has the representation F(I) f h(t (A.17)

(A.18)

Hom (h"TM, f* TM)

F■bf h :

—>

F4'f h(t,

= V2 exP exPf-Rit) (exPh(t) C)) -10 V2 exP(C) •

Thus, according to the fundamental Lemma A.5,

(A.19)

S

is a smooth map.

G(8(h*TM);(S(f*TM))

(Oh) 1-0

(F4vh , s) = D4:) f h * (8)

216

APPENDIX A. CURVE SPACES AND BANACH BUNDLES

Remark Similar to the proof of this fundamental lemma, we may verify also that : D —■ Horn (TM, TM) gives rise to a smooth map 6(0) : 6(0h) —*

(A.20)

G(6(11*TM);6(h*TM))

1—* 6* (0 Now let T: Vecc.(11i) that

s) .

Ban be any section functor satisfying the condition

T. : 6 (Horn (e, q)) A 1—* (si—* A • s) yields a continuous map for each two (A.21)

E Vec(Pri), that is, in short terms,

6(Hom) C G(l; T) .

Referring to the estimate (A.1) in the auxiliary proposition A.2, we notice that T = Li fulfills this condition with respect to 6= and thus in particular

cl

with respect to 6= 14 2 due to (A.3). Given a fixed smooth section so E cr (O h), we are provided with a smooth map as in the proof of the fundamentarlemma, 19* : 6(0h )

(A.22)

6 (Hom (h*TM, f *TM)) 1—)• F4 fh0 —

F(Dfho so •

It holds that 6* 019.(5)

= 61(FID fh0 S — F4) fh0 SO)

= 6 * (F(Df h0 S) — 6.(113 f h0 So) E

12(6(h*TM); 6 (f * TM)) -

Thus, property (A.21) of the section functor T yields the smooth mapping

(A.23)

6(0h) —* G(T(h*Tm);T(f*TM)) S

1—*

T. (FIDA0 s — F(I) fh°

SO) •

Since T.(F ■Dfhoso) E G(T(h*TM);T(f*TM)) is already well-defined according to the section functor property, we obtain the smooth map (A.24)

T1 : e(Oh ) —* s

Ger(h*TM);T(f*TM))

Ta (F,kf h0 5)

in analogy with G(6(h 1 TM); 6(1 'TM)) .

After these preparations we are able to prove the following

A.2. BANACH BUNDLES ON

P,Ii2,1,(R, M)

217

Theorem 11 The section functorT satisfying condition (A.21) admits a unique extension to the continuous vector bundles of the shape g*TM, g E 7,1:121 (R, M) C C?,11 (1,M); x,y E M . Moreover, T

(P1:12i *TM) = U

T(g*TM)

gEr(R,M)

is a Banach vector bundle on

P1:72,(R,M).

Proof. Let a, 0 E CL(R,M) be given, satisfying

0 = expo, CE C° (a*D) c C° (a* TM) . We consequently define Joa E C° (Horn (a''TM, 0*TM)) by (A.25)

J,3„(t) • v(t) =- V2 exP (e(t)) • v(t) .

Thus, Jpc,(t) is a toplinear isomorphism for each t E. Now let us consider charts based on the supporting curves h, f E CZ (IR- , M) as in the proof of Theorem 10. Formula (A.18) applied to s E 6(0h) and v E 6(h*TM) leads to the identity V2 exp(Ilfh o s) o (F(1)fh0 .9)- y = V2 eXP(8)• y ,

that is (A.26)

6.(Fcl)fh0 s) = 6. ((./9 1) -1 0 Jo),

g = exph s E 7)xl:y2 (1R, M) .

We therefore define the fibre at g = exph s as T(g*TM) = {Jgh•viv ET(h*TM)} .

This is independent of the choice of h, because (A.26) implies the equation (A.27)

.19 f 0 6,(F
Thus, referring to (A.24), the Banach space topology on T(g*TM) is defined in a unique way independent of the underlying curve h. The smoothness of the map T. : 6 (0h)

—■

G(T(h*TM); T(f*TM))

s

1-4

T. (F(I)fh0

.9)

due to (A.24) implies that the maps Jgh :T(h * TM) —■ T(g*TM)

218

APPENDIX A. CURVE SPACES AND BANACH BUNDLES

( provide us with a well-defined local trivialization of the bundle Tp transition maps. together with smooth Since the transition maps due to (A.27) comply with the identity

(A.28)

f) 1 0 Jgh 8*(F4) f h 0 45) = DI) fh * (s)

with respect to the special section functor T = S, the Banakh bundle (pl. :1127m) =

U 6(h*TM) tieP1:2„(k,m)

represents exactly the tangent bundle of the Banach manifold a consequence we obtain

M). As

Corollary A.8 T1'x1:y2 (R,M) 4'2 (P1'12,*TM) . Remark A more detailed analysis of the exponential map, its derivatives V1 exp and V2 exp and of the Levi-Civita connection would yield the possibility of discussing the associated exponential map expT on the Riemannian manifold TM. As a consequence, the functorial behaviour of P. 2 (R, .) as stated in Corollary A.3 applied to the projection map 7 : TM M would enable us to find a representation of the tangent bundle TPx1 ;y2 in the following form:

T77,111 (1a, M) = Pesly (R, TM), 7)k2oy (R, TM)17- o

{

T,1).1 :y2 (R, M)

-y },

pxl :y2 .

pEl. , y2 as According to this representation we may grasp the tangent space the space of the '4' 2-vector fields along the 4' 2-curve -y' equipped with a Banach space topology.

Theorem 12 Let X E Cc° (TM) be a smooth vector field on M satisfying X(x) = X(y) = O. Then the mapping CZy (fit , M) 3 7 I—, 7+X 0-y E Cc° ey*T M)

can be extended to a smooth section in the Banach bundle Li(P1 4,2*TM):

r

21 :1,2(R ,

m)

Li(rl :v2.Tm-) +Xo-yE Liey*TM) .

'Y

Proof. It is sufficient to verify the smoothness of this map with respect to the local representation by a chart together with the overlying local trivialization. We thus consider J4' 2 (Uh) x at (h*TM) (exp h

77)

Li(P1:1,2 *TM) 1 4,2 (uh) v2

exP()

17 •

A.2. BANACH BUNDLES ON 'T) ,2y (R, M) The local representation of the section

219

r consequently gets the shape

Li(h*D)

rtriv,h : 4 2 (h*D)

(A.29)

V2 exP() (Vi exP() • it +V2 exP() • V + X(exPtt )) •

We may also express this as (A.30)

Ttriv,h() =

e(e) • ii + V + V2 exP(e)-1. (X

o exPh)() -

As we already noticed in (A.20), the mapping

42 3

e ( ) E L(4 2 (h* TM); 4 2 (h*T M))

is smooth, therefore in particular the mapping (A.31)

e() h. E Hille(h*TM)

for

it E Cr(h*TM)

is also. We furthermore observe that

(A.32)

41'2

Vte E

is continuously linear and thus also smooth. Considering now the open zero section neighbourhood 0 = hD C hsTM, we are led to the smooth bundle map (A.33)

f : 0 -4 h*TM

f (v) = V2 exp(v) -1 . X (exph v) satisfying the condition f (0 ±c„) ) = 0+0„ due to the assumption X(x) = X(y) = O. Hence, the fundamental Lemma A.5 guarantees the smoothness of the map Thit ,2 (h * D) 14,2 (h i 7-, m) Piecing all this together proves the assertion. Corollary A.9 Given a smooth Morse function f on M, the mapping F - La(P T M), x , y E Crit f -

.§+Vfos

describes a smooth section in the specified L 2 -Banach bundle. The local representation at 7 E C , is of the type Fl0c,1 (7 * D) Li(7 * M) g)(t) = V(t) g(t, . (t)) , where g : Ft x -y*D 7*TM is smooth and fibre respecting, satisfies the identity g(±co, 0) = 0 and is endowed with the asymptotic fibre derivatives D2g(±oo, 0) which are conjugated linear operators of the Hessians H2 f (x) and H2 f (y), respectively.

Appendix B

The Geometric Boundary Operator Referring to the brief historical outline in the introduction, we notice that another, geometric definition of the characteristic signs for isolated a-trajectories of the time-independent gradient flow was given there: Definition B.1 Let f be a smooth Morse function on M together with the critical points x, y E Crit f. We assume that the Morse-Smale condition is fulfilled, so that

(B. 1 )M1,y "''' Wu (x) (fil478 (Y) = M(x, y) represents a 1.1(x) — ti(y)-dimensional manifold consisting of parametrized gradient trajectories. Let all unstable manifolds Wu(x) for x E Crit f be endowed with arbitrary and noncanonically fixed orientations. Provided with the additional assumption that M is orientable and oriented, we thus obtain immediately induced orientations for the associated, codimensional stable manifolds Ws(x), x E Crit f. We now consider the short exact sequence associated to a non-critical point p E M(X, y) (B.2) where

0

Tp M(x , y)

Tp 141 1 ' (x) 6 Tp W5 (y)

Tp M

0

i( u) = (u, —u) j (v , w) = v + w .

We note that exactness is essentially due to the transversal intersection as indicated in (B..1). Starting from the given orientations on Tp Wu(x), T p W .9 (y) and TM, we are supplied with an orientation on the connected component M(x,y)(;) of p, which is induced by this short exact sequence. Thus we obtain

222

APPENDIX B. THE GEOMETRIC BOUNDARY OPERATOR

orientations og ,,,[u] of the operator classes [u, Du ] E A in a geometrical way for orientable manifolds M. These operator classes belong to the trajectories

u = uxy E (x, y) E X = { (x, y) E Critkf x Critif k >1, M(x , y)

0}

Consequently, the comparison of these geometric orientations with the canonical orientation for isolated trajectories by the free R-action from time-shifting provides the characteristic signs of the geometric boundary operator. Thus the sum of all characteristic signs for fixed endpoints x, y is nothing more than the intersection number yielded by the transversal intersection

(B.3)

Wu(x) rh Ws(y) rt Ma

within the level surface Ma associated to a regular value f (y) < a < f (x). This level surface is oriented canonically by the normal field —V f .

Theorem 13 The noncanonical geometric orientation ogeom of the operator classes [un], (x, y) E X can be extended to a coherent orientation u E CA. Proof. Let us mention first that it suffices to consider the equivalence classes {u} E CA in the sense of (4.18) from Section 4.1.4. We further notice that the

transformation group {±1 Cri }t f on the set of these geometric orientations acts in the same way on en as it is expressed in (4.19). Therefore it is sufficient to verify the condition of coherence (B.4)

Ogeorn [Uxt,}

# Ogeom[Uvz] = Ogeom[ttxy#ptlyz]

for the geometric orientation. This will be reduced step by step to a simpler relation for the linear gluing version from Section 3.1.2. Throughout the following technical discussions we will refer to the notations from Section 2.2 on Fredholm theory. Definition B.2 We define Edtrliavg , the so-called set of Etriv -operators in ordered diagonal form, as the subset of Etriv comprising the operators K = a + D, for which D E A has the following shape: at

, An), Ai E CID (R,R) > 0, for 1 i n where Ai (±oo) = A < 0 , for n kt(A ±) < i n

D = diag(Ai,

1

Lemma B.3 Let us assume a KA c

Et riv with A E A smooth and asymptotically constant, i.e. A(t) = const ± for I ti T. Then there exists a conjugation transformation by an asymptotically constant C E Cc° (— GL(n,R)), so that k = C-1 KC E Eci„iiayg has ordered diagonal form.

223 Proof The first step is to construct a C± E GL+ (n, R), such that we obtain the representation (B.5) CI 1 A+ C + = diag(AP,... , with ordered eigenvalues as specified in Definition B.2. We consequently choose an asymptotically constant curve C E C (ft, GL+ (n,R)) with C(±oo) = as in the conclusion of Lemma 2.15. Thus, without loss of generality, A is already endowed with ends in ordered diagonal form, AI = diag(4,...,A„± ) .

(B.6)

In the following step we consider the ordinary linear differential equation

.qt) = -A(t) s(t) and the associated evolution operator

U(-) = U(-T,

E

C" (R, GL + (n, R))

with respect to

1.I(t) = -A(t) • U(t), U(-T) =1 .

(B.7) The formula

U(t)

exp

f

A(T)dT)

—T

together with identity (B.6) yields the representations e—A,--(t +7')

(B.8)

for t -T ,

U(t) e —AC.(t+T) )

(B.9) U(t) =

for t2.- T

U(T) e -A;;.(t- T)

Then we define C E C' (R, GL + (n, R)) as follows: 1

13-(t+T)-1 +(1 - (3- (t + T)) - U(t) ' C(t)=J

U(t), U(T) •

lAs to

[ +(1

—

t

-T ,

-Tt“ ' ,

)3+ (t - T) • 11 0+(t T)) • U(T) - qi(t)] , t?" 7' • —

the definition of 'Ell, see Definition 2.47.

224

APPENDIX B. THE GEOMETRIC BOUNDARY OPERATOR

Straightforward computation using identities (B.8) and (B.9) together with the formula

C-1 KC =

0

where A is already given in asymptotically constant diagonal form, shows that conjugation by C yields the diagonalization as described in the assertion. D Corollary BA Let K E Etriv be surjective and asymptotically constant. Due to Lemma B.3 we may assume without loss of generality that K is given in ordered diagonal form. Then the kernel of K is generated by the curves y1,..., v o (K-)_ p(K+) satisfying 2 VI

= Crei+7,_ 0 (K-),

, (t) =

c- e- AZt, ,

c+ e

c.i E t

-T

t T

, ci(t) > 0 always,

-

1 i

The case where we cannot necessarily assume an asymptotical constant operators can be elucidated by the following Lemma B.5 Given K = fi + A E Etriv,

A

E C i (E,End(Rn )) such that

A(t) is self-adjoint for all Iti > T for some T> 0, the following asymptotical property holds for the solutions v E ker K C H 1,2 (R, : lim v(t) = e± ezist and satisfy A ±e± = with A+ > 0, A -
all t E

b(t) = A(t) - v(t), t E R . li v (toll

We denote by e(t) = t E R, the curve e E C(R, Sn -1 ). Straightforward computation shows that e satisfies the ordinary differential equation on Sn -1 C

(B.10)

e(t) = pre_L (t) 0 A(t) • e(t) ,

where el(t) = Te(t)Sn -1 C Rn denotes the subspaces perpendicular to e(t) and preL the orthogonal projection onto the tangent space el. Due to the self-adjointness of A(t) for all t E Fl this differential equation can be restated as the equation associated to the time-dependent gradient flow of f: R x Sn -1

R, ft(e) = (A(t)e ,

.

2ei denotes the i-th vector of the canonical standard basis of R",

225 Similar to the calculations in the compactness analysis for time-dependent trajectories we obtain

a

d (B.11) (é(s),é(s)) = (Vf s (e(s)) , é(s)) = — dt ( f t(e(t)) ) (s)

ft (e(s))

(ii(s)e(s), e(s)). Thus, Lemma 2.2 applied to A E ft (e(s)) End(Rn)) implies that where

(R,

— f t (e(s))ds < oo

at _c

exists, and since Sn-1 is compact, t obtain the finite L2-norm

f (e(t)) is bounded. Altogether we

co

L (é(s), é(s))ds [

(B.12)

< oo

and consequently t urn é(t) = 0. Due to the compactness of S71-1 the assertion

0

now follows.

Continuation of the proof of Theorem 13. Given (uzy , uyz) E A4-; y X Mif,, z and (x, y), (y, z) E X, we choose chart neighbourhoods tin and Liyz on the

Banach manifolds PPy2 aund /17,! at asymptotically constant smooth curves hxy and hyz , i.e. e E C") (R, h4T

uzy = exPn xi,

In these local coordinates, the focal Fredholm map Ft,

.

1+ V f

is expressed as

= OW • it + V + V2 exp(6) -1 - (VI o exph 6) .

We denote the differential of Fu. at 6 E H 2 (h! . TM) by Du... (e). In the following analysis we argue that we may assume without loss of generality that = V t + A( ) is asymptotically constant, which generally is only given if 6 has compact support. Defining a cut-off function OT E c—(t, [0,1 ] ) for T E 10,1] by

i3o a- 1, (3,r =

1,

It'

0,

iti otherwise

monotone,

+, - 1

{

and the continuous path

OT :

[0,1 ] T

-■ Cœ(17, h!..T M)

13T

for T > 0

APPENDIX B. THE GEOMETRIC BOUNDARY OPERATOR

226

we obtain the continuous path of Fredholm operators T Du (&) with Du = Du (6). We find a To > 0 such that D u (6, ) is surjective for all 0 T To . Thus, the geometric orientation ogeom [u...] of ker Du induces a unique orientation of ker D which can be determined by explicit analysis of the asymptotic behaviour of the solutions generating the kernel in the same way as for the original operator which is not asymptotically constant. This follows from the given continuation argument for T 0 together with Lemma B.5 by using the fact that 6 , is R-differentiable for all T E [0,7'0 ]. Referring to Lemma 3.13 we notice that, concerning the verification of the coherence condition, it suffices to analyse the local situation at uxy , v uxy rpuyz by means of local coordinates and trivializations. To sum up, we may start without loss of generality from asymptotically constant and surjective operators Now let Ts Wu(x), T v Wu(y) and Tz Wu(z) be endowed with fixed orientations. Using the trivializations of u*xv TM and u;z Tilf , which coincide at TM and transform the operators D Duv.. into ordered diagonal form, we are led to the orientations ez , A... A ezu( .,

(B.13) e3 ,

on

span (en_ m (x)+1, - , en) = Tx Wu(z) ,

A ... A

e3H(v) on

span (en _m(y) +1 ,... , en )

Ty lVu(y) ,

ek l A... A

ek i, co on

span (en_ i,(t)+1, - • • , en)

TzWu(z) .

According to Corollary B.4 the kernels ker Dtitriv and ker Du„,trtv are generated by . , vii(x )_ 4 (y ), V = ct. et+n---st(s), c >0, 1

i

ii(x) —

and W11 • • •

Wi(y)-p(z),

103 = C/3 ' ej+n-A(Y)

> 0, 1

tt(y) —

j

respectively. Then the respective geometric orientations (B.14)

ogeorn[unitriv =Ei • Vi A • • - A Vi (x )_( y ),

E {±1}

ogeom[uyzitriv = f2• wi A • • • A

E {±1}

are determined uniquely by

(B.15)

ei, A

A et w . ) = ogeomluxyltriv A ei,

ei, A

... A eim(y) = ogeom [Uyz]triv A ... A

ek,()z

due to Definition B.1. It likewise holds that (B.16)

et , A ... A

= ogeozz,[uz i,#o° uy zitriv A ek, A

.. • A

ek, (z) •

227

Thus, it remains merely to analyse the linear gluing version Duzy,triv (B.17)

#p

ker

x ker D„,t riv 1-4

(1 ()

and

(Du r ,,triv#Ppu y ,triv)

P)

Pr0:11121"

to verify that the orientation Ifr (v#0)

A ... A (vi,(x)_ 0 ( 9)#p0) A (0#pw1) A ... A (0#p wo (y)_ 0 ( z ))

p -yz triv • coincides with the geometric orientation ogeomrn-xy 44°7L

Given the surjective operators Dutriv = ,triv — + Av z E Etriv in ordered diagonal form

A

Axy

E

Etriv

and

Avz = diag (vi , • • • , vn)

with we construct the following [0,1) 3 (B.18)

Alx-y

(1

,n,

i = 1,

+11;, E Etriv C F(H1,2, L2),

± ATsy ,

7

v,

homotopies:

..., An ) 7) •diag diag (xi , • , An- A (y), A n+_ 1.4 )+1 , • • • ,

(1-

AT

A ,-1- =

diag

• diag (v1, ,

, Vn) vn-A(0+1, • • • , vn)

Obviously, both homotopies A + Aix'y and g+Avr z run continuously through surjective operators from Et r i v , , which respectively have constant kernels 0 Ot

and

ker (— +

A'

0 ker (— +

AT

at

)=

span (vi,

Vz ) = span (WI,

in xy ,1 triv and ogeom [uy zIi triv remain conThus, the geometric orientations ogeon', stant during these homotopies. Moreover these homotopies are compatible (A rxy #pA yr z) with the linear gluing version (B.17). This means that 7 1—> describes a continuous homotopy through surjective operators, where p maxi- E [co] Po (D,T , v DuT y . ,triv) may be chosen as independent of 7. During this homotopy the kernel ker (g+A4#,A; z ) is generally not kept invariant, but the orientations ogeom [Uxy eptiyz]triv and Ogeorn [Uxy] triv #0 geo m [1Ly 1 triv remain fixed by reasons of continuity. Due to the construction it holds that c1f2. vi, p A ... A v,( x)_ i,(0, p A tvi,_ pA ... A w o ( y )_( z ),_ p A ek, A ... A ekm(.. )

A ... A

.

228

APPENDIX B. THE GEOMETRIC BOUNDARY OPERATOR

Thus, the identity (B.19)

ogeom

#p °11yz]triv = Og eorn [Uxy] triv # 0geom [Uy,r]triv

follows from the explicit representation of the linear gluing version #p in (B.17) for 7- = 1: vi #p0 = v, pw = wi,p,,

i = 1, . . . , p(x) — p(y) =

gY) 1-1(z)

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99-114. [C]

C. C. Conley, Isolated invariant sets and the Morse index, CBMS Reg. Conf. Series in Math. 38 (1978), A.M.S., Providence, R.I.

[C-Z ]

C. C. Conley and E. Zehnder, Morse—type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207-253.

[Cou]

R. Courant, Dirichlet 's principle, conformal mappings and mimimal surfaces, Interscience, New York, 1950, reprinted: Springer, New

York—Heidelberg—Berlin, 1977. [Don]

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Index ■••

admissible covariant derivation, 39 pair of manifolds, 11, 163, 180, 180, 183, 184, 190, 194, 206 pair of trivializations, 115, 120, 159, 163, 201 Arz,e1a-Ascoli, theorem of, 57 associativity, 68, 113, 123 asymptotically constant, 71, 107, 122 Banach bundle, 207, 217

manifold, 14, 22, 24, 28, 42, 43, 46, 70, 100, 207, 212, 214, 218, 225 boundary operator, 133, 202, 221 canonical boundary operator, 133

homomorphism, 151 identification, 175, 183 isomorphism, 133, 140, 152, 153 orientation, 142, 147 chain

complex, 136, 180, 184, 186, 202 homomorphism, 141, 203 homotopy, 144 homotopy operator, 187 isomorphism, 206 map, 163, 169 characteristic class, 183 characteristic sign, 133, 145, 155, 200, 204, 221, 222 cobordism, 16, 70, 137 equivalence, 52, 95, 142, 147

of trajectory spaces, 95 oriented, 103 coboundary operator, 200 cochain complex, 199 coercivity, 10, 55, 63-66, 164, 166, 181, 182, 191, 200 coherent orientation, 15, 129, 157, 158, 161, 169, 200, 203, 222 cohomology theory, 199 compactness, 15, 51 -gluing cobordism, 70 result, 63, 200 complementarity, 95, 103 conjugated self-adjoint, 29, 41, 130, 156 connecting orbit, 136, 161 continuation principle, 6, 9 contraction principle, 84 correction term, 72, 80, 85, 87, 99 covariant derivation, 39 Fredholm admissible, 39, 114 induced, 40 determinant bundle, 16, 103, 107, 108, 113,

116, 123, 158 space, 104, 111

diffeomorphism type, 95, 146 dimension axiom, 121, 156, 191, 193 elliptic regularity, 26, 57 embedding closed, 163, 170, 174, 177, 179,

188 diagonal, 205

INDEX

233

for trajectory spaces, 69 graph-, 177 of gluing, 126 of trajectory spaces, 97 equivalence of broken trajectories, 137, 148 of coherent orientations, 222 of Fredholm operators, 35, 107,

111, 129, 154, 156 of Fredholm operators along curves, 115 of homology classes, 10 of one-sided operators, 158 of operators, 159 of simply broken trajectories,

95 evaluation map, 53 excision axiom, 190 exponential map, 211, 213, 215, 218 factorization, 176, 177, 177, 180 five lemma, 187 Floer homology, 6, 14, 49, 52, 76 Fredholm class, 133 index, 15, 35, 38, 47, 49, 108 local, 41 map, 41, 42, 43, 47, 103, 225 operator, 15, 16, 22, 30, 34, 40,

42, 47, 74-76, 80, 101, 105107, 126, 133, 157, 158 generic, 28, 42, 44, 49, 50 geometric boundary operator, 222 geometrical convergence, 56, 81 gluing, 15, 68, 123 and orientation, 113 for one-sided operators, 158 of A-parametrized trajectories,

127 of asymptotically constant curves, 71 of Fredholm operators, 108, 110

of trajectories, 80, 87, 123, 138,

147 of trivializations, 122 pre-, 70, 71, 74, 88, 97, 110,

125 unparametrized, 88 gradient flow, 86 negative, 14, 26, 53, 55, 70, 80, 156, 159, 168, 171, 173, 181, 192, 200 positive, 156, 159, 200 time-independent, 174 Hessian, 1, 26, 29, 78, 86, 156, 162,

219 homology group absolute, 163 relative, 163, 184, 190 isomorphism, 170 canonical, 172

homotopy, 9 A-, 9, 50, 100, 126, 187 finite, 46 invariance, 123, 163, 190 lemma, 171, 174, 176, 178, 180,

188 operator, 10, 15, 22, 144, 187 regular, 46, 47, 49, 63, 144 trajectory, 63, 98 identification canonical, 172, 202 identification process, 153, 161, 171,

187, 200 index Fredholm, 15, 47, 105, 108 Morse, 15, 62, 145 intersection number, 3, 103, 136

Kiinneth formula, 204 Levi-Civita connection, 214, 218 long exact sequence, 180, 184-186, 187, 190

234

INDEX

Möbius band, 121, 155 manifold of maps, 22 Morse cohomology, 200 complex, 6, 8, 11, 133, 135, 136,

163, 167, 169, 171, 180, 184, 206 relative, 184, 187, 206 function, 1, 10, 65, 133, 135, 151, 163, 165, 167, 169, 181, 184, 201, 202, 219 relative, 11, 183, 184, 190, 206 homology, 2, 10, 63, 153, 167, 180 group, 162, 177, 184 relative, 190 homotopy, 21, 63, 64, 96, 100, 140, 143, 151, 152, 172, 201, 204-206 regular, 185 relative, 185, 187, 188 index, 1, 29, 62, 135, 167, 191, 200, 201, 204 relative, 3, 6, 9, 10, 15, 30, 35, 38, 51, 67, 95, 134, 136, 145, 174 inequalities, 2 lemma, 192 theory, 184 relative, 15 Morse-Smale condition, 3, 15, 28, 42, 44, 221 mountain-pass lemma, 192 normal bundle, 12, 70, 72, 75, 76,

182 normalization, 194

canonical, 134, 138, 142, 147, 163, 222 coherent, 103, 129, 131, 135, 137, 151, 153, 155, 157, 158, 200 geometric, 222, 226 of Fredholm operators, 103, 107 of trajectory manifolds, 134 Palais-Smale condition, 55, 62 Poincaré duality, 13, 114, 156, 200 product cross, 205

homological, 204 cup, 13, 205 tensor, 202 projection, 175, 177 quotient complex, 180, 184

regularity, 49 reparametrization, 52, 56, 64, 89, 94 Sard-Smale, theorem of, 41, 44 section functor, 24, 207 shifting-invariance, 134 short exact sequence, 184, 186 Sobolev embedding, 24 spectral flow, 15, 30, 35, 78 splitting-up, 15, 61 stable manifold, 159, 161, 192, 221 steep, 181, 182, 185, 188 tangent bundle, 218 Thom isomorphism, 183, 206 Tietze, theorem of, 164

time-shifting, 9, 52, 71, 96, 122, 138, 222 trajectory )-parametrized, 10, 50, 67, 144 h , 63, 98 broken, 15, 51, 56, 63, 76, 81, 88, 148 -

one-sided operator, 157 orbit space, 54 orientation, 112 and gluing, 113

gradient, 28

homotopy, 98

INDEX

235

isolated, 4, 52, 133, 141, 145, 153, 162, 173, 191, 193, 200, 203, 204, 221, 222 manifold, 21, 25, 52, 67, 133 mixed broken, 96, 142 parametrized, 221 simply broken, 52, 69, 93, 95, 137 space, 8, 15, 21, 28, 49, 51

)-parametrized, 22 time-dependent, 22 time-independent, 21 unparametrized, 54, 88

time-dependent, 9, 26, 28 time-independent, 52 unparametrized, 52, 54 , 55, 69, 88

transversal, 14, 42, 54 transversality, 28, 42, 49

tubular neighbourhood, 205 uniqueness category, 194, 198 unstable manifold, 159, 162, 221 vector bundle, 183 weak convergence, 56, 61, 64, 66, 69, 93, 148