Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topolgy

Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology

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Series II: Mathematics, Physics and Chemistry – Vol. 216

Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology edited by

Paul Biran Tel-Aviv University, Israel

Octav Cornea University of Montreal, QC, Canada and

Franç ois Lalonde University of Montreal, QC, Canada

Published in cooperation with NATO Public Diplomacy Division

Proceeding of the NATO Advanced Study Institute on Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology Montré al, Canada July 2004

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CONTENTS

Preface

xi

Contributors

xiii

A. Abbondandolo and P. Majer/ Lectures on the Morse Complex for Infinite-Dimensional Manifolds 1 A few facts from hyperbolic dynamics 1.1 Adapted norms 1.2 Linear stable and unstable spaces of an asymptotically hyperbolic path 1.3 Morse vector fields 1.4 Local dynamics near a hyperbolic rest point 1.5 Local stable and unstable manifolds 1.6 The Grobman – Hartman linearization theorem 1.7 Global stable and unstable manifolds 2 The Morse complex in the case of finite Morse indices 2.1 The Palais – Smale condition 2.2 The Morse – Smale condition 2.3 The assumptions 2.4 Forward compactness 2.5 Consequences of compactness and transversality 2.6 Cellular filtrations 2.7 The Morse complex 2.8 Representation of ∂∗ in terms of intersection numbers 2.9 How to remove the assumption (A8) 2.10 Morse functions on Hilbert manifolds 2.11 Basic results in transversality theory 2.12 Genericity of the Morse – Smale condition 2.13 Invariance of the Morse complex 3 The Morse complex in the case of infinite Morse indices 3.1 The program 3.2 Fredholm pairs and compact perturbations of linear subspaces 3.3 Finite-dimensional intersections

v

1 2 2 4 8 9 11 12 19 21 21 22 22 23 25 27 28 33 37 38 40 42 47 52 52 54 54

vi

CONTENTS

3.4 Essential subbundles 3.5 Orientations 3.6 Compactness 3.7 Two-dimensional intersections 3.8 The Morse complex Bibliographical note

57 59 61 65 67 69

A. Abbondandolo and M. Schwarz/ Notes on Floer Homology and Loop Space Homology 75 1 Introduction 75 2 Main result 77 2.1 Loop space homology 77 2.2 Floer homology for the cotangent bundle 80 3 Ring structures and ring-homomorphisms 85 3.1 The pair-of-pants product 85 3.2 The ring homomorphisms between free loop space Floer homology and based loop space Floer homology 89 and classical homology 4 Morse-homology on the loop spaces ΛQ and ΩQ, and the 95 isomorphism 5 Products in Morse-homology 102 5.1 Ring isomorphism between Morse homology and 105 Floer homology J.-F. Barraud and O. Cornea/ Homotopical Dynamics in Symplectic Topology 1 Introduction 2 Elements of Morse theory 2.1 Connecting manifolds 2.2 Operations 3 Applications to symplectic topology 3.1 Bounded orbits 3.2 Detection of pseudoholomorphic strips and Hofer’s norm

109 109 110 113 125 127 127 130

R. L. Cohen/ Morse Theory, Graphs, and String Topology 1 Graphs, Morse theory, and cohomology operations 2 String topology 3 A Morse theoretic view of string topology 4 Cylindrical holomorphic curves in the cotangent bundle

149 152 165 173 178

M. Farber/ Topology of Robot Motion Planning 1 Introduction 2 First examples of configuration spaces 3 Varieties of polygonal linkages

185 185 186 189

CONTENTS

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

3.1 Short and long subsets 3.2 Poincar´e polynomial of M(a) Universality theorems for configuration spaces A remark about configuration spaces in robotics The motion planning problem Tame motion planning algorithms The Schwarz genus The second notion of topological complexity Homotopy invariance Order of instability of a motion planning algorithm Random motion planning algorithms Equality theorem An upper bound for TC(X) A cohomological lower bound for TC(X) Examples Simultaneous control of many systems Another inequality relating TC(X) to the usual category Topological complexity of bouquets A general recipe to construct a motion planning algorithm How difficult is to avoid collisions in Rm ? The case m = 2 
TC F(Rm , n) in the case m ≥ 3 odd Shade Illuminating the complement of the braid arrangement A quadratic motion planning algorithm in F(Rm , n) Configuration spaces of graphs Motion planning in projective spaces Nonsingular maps TC(RPn ) and the immersion problem Some open problems

K. Fukaya/ Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology 1 Introduction 2 Lagrangian submanifold of Cn 3 Perturbing Cauchy – Riemann equation 4 Maslov index of Lagrangian submanifold with vanishing second Betti number 5 Floer homology and a spectral sequence 6 Homology of loop space and Chas – Sullivan bracket 7 Iterated integral and Gerstenhaber bracket 8 A∞ deformation of de Rham complex

vii 190 190 194 195 195 198 199 200 201 201 202 203 207 207 208 209 210 211 212 213 214 216 217 219 220 221 222 224 226 228 231 231 232 235 240 243 246 251 254

viii

CONTENTS

9 10 11 12 13 14

S 1 equivariant homology of loop space and cyclic A∞ algebra L∞ structure on H(S 1 × S n ; Q) Lagrangian submanifolds of C3 Aspherical Lagrangian submanifolds Lagrangian submanifolds homotopy equivalent to S 1 × S 2m Lagrangian submanifolds of CPn

M. Izydorek/ The LS-Index: A Survey 1 Introduction 2 The LS-index 2.1 Basic definitions and facts 2.2 Spectra 2.3 The LS-index 3 Cohomology of spectra 4 Attractors, repellers and Morse decompositions 5 Equivariant LS-flows and the G-LS-index 5.1 Symmetries 5.2 Isolating neighbourhoods and the equivariant LS-index 6 Applications 6.1 A general setting 6.2 Applications of the LS-index 6.3 Applications of the cohomological LS-index 6.4 Applications of the equivariant LS-index Y.-G. Oh/ Lectures on Floer Theory and Spectral Invariants of Hamiltonian Flows 1 Introduction 2 The free loop space and the action functional 2.1 The free loop space and the S 1 -action in general 2.2 The free loop space of symplectic manifolds 2.3 The Novikov covering 2.4 Perturbed action functionals and their action spectra 2.5 The L2 -gradient flow and perturbed Cauchy – Riemann equations 2.6 Comparison of two Cauchy – Riemann equations 3 Floer complex and the Novikov ring 3.1 Novikov – Floer chains and the Novikov ring 3.2 Definition of the Floer boundary map 3.3 Definition of the Floer chain map 3.4 Semi-positivity and transversality 3.5 Composition law of Floer’s chain maps 4 Energy estimates and Hofer’s geometry 4.1 Energy estimates and the action level changes

259 260 265 266 271 272 277 277 282 282 285 288 290 293 296 296 296 303 303 305 309 312 321 321 326 326 327 328 330 332 336 338 338 341 346 347 348 352 352

CONTENTS

5

6

7

8

9

10 A

4.2 Energy estimates and Hofer’s norm 4.3 Level changes of Floer chains under the homotopy 4.4 The -regularity type invariants Definition of spectral invariants and their axioms 5.1 Floer complex of a small Morse function 5.2 Definition of spectral invariants 5.3 Axioms of spectral invariants The spectrality axiom 6.1 A consequence of the nondegenerate spectrality axiom 6.2 Spectrality axiom for the rational case 6.3 Spectrality for the irrational case Pants product and the triangle inequality 7.1 Quantum cohomology in the chain level 7.2 Grading convention 7.3 Hamiltonian fibrations and the pants product 7.4 Proof of the triangle inequality Spectral norm of Hamiltonian diffeomorphisms 8.1 Construction of the spectral norm 8.2 The -regularity theorem and its consequences 8.3 Proof of nondegeneracy Applications to Hofer geometry of Ham(M, ω) 9.1 Quasi-autonomous Hamiltonians and the minimality conjecture 9.2 Length minimizing criterion via ρ(H; 1) 9.3 Canonical fundamental Floer cycles 9.4 The case of autonomous Hamiltonians Remarks on the transversality for general (M, ω) Proof of the index formula

L. Polterovich/ Floer Homology, Dynamics and Groups 1 Hamiltonian actions of finitely generated groups 1.1 The group of Hamiltonian diffeomorphisms 1.2 The no-torsion theorem 1.3 Distortion in normed groups 1.4 The No-Distortion Theorem 1.5 The Zimmer program 2 Floer theory in action 2.1 A brief sketch of Floer theory 2.2 Width and torsion 2.3 A geometry on Ham(M, ω) 2.4 Width and distortion 2.5 More remarks on the Zimmer program

ix 355 358 359 361 361 362 365 367 368 370 374 377 377 380 382 385 386 386 389 394 399 399 401 403 404 406 408 417 417 417 418 420 421 422 423 423 425 425 426 426

x

CONTENTS

3

The Calabi quasi-morphism and related topics 3.1 Extending the Calabi homomorphism 3.2 Introducing quasi-morphisms 3.3 Quasi-morphisms on Ham(M, ω) 3.4 Distortion in Hofer’s norm on Ham(M, ω) 3.5 Existence and uniqueness of Calabi quasi-morphisms 3.6 “Hyperbolic” features of Ham(M, ω)? 3.7 From π1 (M) to Diff 0 (M, Ω)

428 428 429 430 431 433 434 435

C. Viterbo/ Symplectic topology and Hamilton – Jacobi equations 1 Introduction to symplectic geometry and generating functions 1.1 Uniqueness and first symplectic invariants 2 The calculus of critical level sets 2.1 The case of GFQI 2.2 Applications 3 Hamilton – Jacobi equations and generating functions 4 Coupled Hamilton – Jacobi equations

439 439 446 447 450 453 454 456

Index

461

PREFACE

The papers collected in this volume are contributions to the 43rd session of the S´eminaire de math´ematiques sup´erieures (SMS) on “Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology.” This session took place at the Universit´e de Montr´eal in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most significant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are reflected in the present volume: Morse theory and related techniques in infinite dimensional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian diffeomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all participants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane B´elanger, our administrative assistant, for her help with the organization and Mr. Andr´e Montpetit, our technical editor, for his help in the preparation of the volume. The ASI was made possible by the financial support from the Scientific and Environmental Affairs Division of NATO, the “Centre de recherches math´ematiques” from Montr´eal and the Universit´e de Montr´eal. We are most grateful to all three organizations. Paul Biran Octav Cornea Franc¸ois Lalonde

xi

CONTRIBUTORS

Alberto Abbondandolo Dipartimento di Matematica Universit`a di Pisa via Buonarroti 2 56127 Pisa Italy [email protected] Jean-Franc¸ois Barraud UFR de Math´ematiques Universit´e de Lille 1 59655 Villeneuve d’Ascq France [email protected] Octav Cornea D´epartement de math´ematiques et statistique Universit´e de Montr´eal Montr´eal, QC H3C 3J7 Canada [email protected]

Kenji Fukaya Department of Mathematics Kyoto University Japan [email protected] Marek Izydorek Faculty of Applied Physics and Mathematics Gda´nsk University of Technology ul. Gabriela Narutowicza 11/12 80-952 Gda´nsk Poland [email protected] Pietro Majer Dipartimento di Matematica Universit`a di Pisa via Buonarroti 2 56127 Pisa Italy [email protected]

Ralph L. Cohen Department of Mathematics Stanford University 450 Serra Mall, Bldg. 380 Stanford, CA 94305-2125 USA [email protected]

Yong-Geun Oh Department of Mathematics University of Wisconsin Madison, WI 53706 USA [email protected]

Michael Farber Department of Mathematics University of Durham Durham HH1 3LE UK [email protected]

Leonid Polterovich School of Mathematics Tel Aviv University Tel Aviv 69978 Israel [email protected]

xiii

xiv

CONTRIBUTORS

Matthias Schwarz Mathematisches Institut Fakult¨at f¨ur Mathematik und Informatik Universit¨at Leipzig Augustusplatz 10/11 04109 Leipzig Germany [email protected]

Claude Viterbo Centre de Math´ematiques Laurent Schwartz ´ Ecole Polytechnique 91128 Palaiseau Cedex France [email protected]

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS ALBERTO ABBONDANDOLO AND PIETRO MAJER Universit`a di Pisa

Abstract. After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the assumption that rest points have finite Morse index. Then we extend these ideas to rest points with infinite Morse index and co-index, by using a suitable subbundle of the tangent bundle of M as a comparison object.

Introduction These lectures consist of three parts. In the first one we review some results about the dynamics of differentiable flows with hyperbolic rest points, in a Banach space setting. In particular, we prove the local stable manifold theorem, the Grobman – Hartman linearization theorem, and we describe the global stable and unstable manifolds in the case of a flow admitting a Lyapunov function. In the second part we study the Morse complex of gradient-like flows on Banach manifolds, assuming that all the rest points have finite Morse index. We introduce this chain complex as the cellular chain complex of a suitable cellular filtration of the underlying manifold M. In particular, the homology of the Morse complex is isomorphic to the singular homology of M (or to the singular  A), in the relative case, in which we consider a grahomology of the pair ( M,  with a positively invariant open set A, and we consider dient like flow on M,  \ A¯ in the construction of the Morse complex). Then the rest points in M = M we describe the chain boundary operator in terms of the intersection numbers of the unstable and stable manifolds of pairs of rest points with index difference equal to 1. Finally, we specialize the analysis to the negative gradient flow of a Morse function on a Riemannian Hilbert manifold. In this case, we prove that the Morse – Smale transversality assumption holds for generic perturbations of the metric, and that the isomorphism class of the Morse complex does not depend on the metric. These results provide an alternative approach to infinite-dimensional Morse theory, as developed by Palais and Smale in the sixties, see Palais (1963) and Smale (1964a; 1964b). 1 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 1–74. © 2006 Springer. Printed in the Netherlands.

2

A. ABBONDANDOLO AND P. MAJER

The third part is an exposition of recent results by the authors (see Abbondandolo and Majer, 2003b) about the Morse complex approach for gradient-like flows whose rest points have infinite Morse index and co-index. The framework is that of a Hilbert manifold M with a fixed infinite-dimensional and -codimensional subbundle V of the tangent bundle. When the gradient-like flow satisfies suitable compatibility conditions with respect to V, each rest point x can be given a relative Morse index m(x, V), and the unstable and stable manifolds of pairs of critical points x, y intersect in submanifolds of finite dimension m(x, V) − m(y, V). The study of the Hilbert Grassmannian, and in particular of the determinant bundle on the space of Fredholm pairs of subspaces of a Hilbert space, allow to prove that these intersections carry coherent orientations. Finally, suitable integrability assumptions on V, together with compactness assumptions on the flow, imply that the above intersections have compact closure in M. These facts allow to define the Morse complex. The first two parts contain fairly detailed proofs of all the statements, most of which — especially in the second part — are folklore results, for which we could not find appropriate reference in the literature. The style of the third part is different: proofs are only sketched, or given in a simplified framework. We refer to Abbondandolo and Majer (2003b) for a more complete presentation. 1. A few facts from hyperbolic dynamics 1.1. ADAPTED NORMS

Let E be a real Banach space. A bounded linear operator L on E is said hyperbolic if its spectrum does not meet the imaginary axis1 : σ(L) ∩ iR = ∅. In this case, the decomposition of the spectrum of L into the disjoint closed subsets σ+ (L) = σ(L) ∩ {Re z > 0} and σ− (L) = σ(L) ∩ {Re z < 0} induces the splitting E = E u ⊕ E s into L-invariant closed linear subspaces, such that σ(L|E u ) = σ+ (L) and σ(L|E s ) = σ− (L), with projectors Pu = χ{Re z>0} (L), Ps = χ{Re z<0} (L). The spaces E u = E u (L) and E s = E s (L) are often called the positive (or unstable) and the negative (or stable) eigenspaces of L (although they may not consist of eigenvectors). An L-adapted norm is an equivalent norm  ·  on E such that: ξ = max{Pu ξ, Ps ξ},

∀ξ ∈ E,

(1)

e−tL ξ ≤ e−λt ξ ∀ξ ∈ E u .

(2)

and there is λ > 0 such that for every t ≥ 0 etL ξ ≤ e−λt ξ ∀ξ ∈ E s ,

1 In the framework of discrete dynamical systems, a hyperbolic operator is a bounded operator whose spectrum does not meet the unit circle. In that context, an operator L satisfying σ(L)∩iR = ∅ should be called infinitesimally hyperbolic.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

3

As a consequence, also e−tL ξ ≥ eλt ξ ∀ξ ∈ E s ,

etL ξ ≥ eλt ξ ∀ξ ∈ E u ,

(3)

for every t ≥ 0. Such an adapted norm exists. Actually, for every λ in the interval ]0, min | Re σ(L)|[ there is a norm  ·  satisfying (1), (2), and (3). The construction is based on the following lemma, applied to L|E s and to −L|E u . LEMMA 1.1. Let L be a bounded operator on the Banach space (E,  · 0 ), and let λ be a real number such that λ > max Re σ(L). Then there exists a norm  ·  on E equivalent to  · 0 such that etL ξ ≤ etλ ξ for every ξ in E and t ≥ 0. Proof. Up to replacing L by L − λI, we may assume that λ = 0, so α := max Re σ(L) is negative, and we must find an equivalent norm  ·  for which etL ξ ≤ ξ, for every ξ ∈ E and t ≥ 0. Still denoting by  · 0 the operator norm induced by  · 0 , the spectral radius formula and the spectral mapping theorem imply lim etL 01/t = max |σ(eL )| = max |eσ(L) | = eα < 1. t→+∞

Therefore, there exists c0 > 0 such that etL 0 ≤ c0 eαt/2 for every t ≥ 0, so  +∞ 2c0 ξ0 ∀ξ ∈ E, etL ξ0 dt ≤ ξ := α 0 defines a norm on E not finer than  · 0 . On the other hand, by compactness e−tL 0 ≤ c1 for every t ∈ [0, 1], so  1 1 etL ξ0 dt ≥ ξ0 ∀ξ ∈ E, ξ ≥ c 1 0 and the norm  ·  is equivalent to  · 0 . Finally, for every t ≥ 0 and ξ ∈ E,  +∞  +∞ tL (s+t)L e ξ0 ds = e sL ξ0 ds ≤ ξ, e ξ = 0

t

concluding the proof.




EXERCISE 1.2. Find an adapted norm for the hyperbolic operator on E = R2 defined by the matrix   −1 µ L= , 0 −1 where µ ∈ R, and draw the corresponding unit ball when µ is large. EXERCISE 1.3. Prove that if L is a normal operator on a Hilbert space H, that is L commutes with its adjoint L∗ , then the Hilbert norm is L-adapted.

4

A. ABBONDANDOLO AND P. MAJER

1.2. LINEAR STABLE AND UNSTABLE SPACES OF AN ASYMPTOTICALLY HYPERBOLIC PATH

Let A: [0, +∞] → L(E) be a continuous path of bounded linear operators on the Banach space E, such that A(+∞) is hyperbolic. Let XA : [0, +∞[ → L(E) be the solution of the linear Cauchy problem  XA (t) = A(t)XA (t), XA (0) = I. EXERCISE 1.4. Prove that XA (t) is an isomorphism for every t, and find a linear Cauchy problem solved by its inverse. The linear subspace of E



WAs = ξ ∈ E lim XA (t)ξ = 0 t→+∞

is said the linear stable space of the asymptotically hyperbolic path A. Similarly, if A: [−∞, 0] → L(E) is a continuous path of operators such that A(−∞) is hyperbolic, the linear unstable space of A is defined as 

WAu = ξ ∈ E lim XA (t)ξ = 0 . t→−∞

EXERCISE 1.5. Prove that if A(t) ≡ L is constant (and hyperbolic), then WAs = E s (L), the negative eigenspace of L, and WAu = E u (L), the positive eigenspace of L. A consequence of the hyperbolicity of A(+∞) is that the linear subspaces WAs
 and WAu are closed and complemented in E, and they are isomorphic to E s A(+∞)
 and to E u A(−∞) , respectively. Indeed, one can prove that if A is close enough to A(+∞) in the L∞ norm, then WAs is the graph of a bounded operator from

 E s A(+∞) to E u A(+∞) . The statement for a general asymptotically hyperbolic path A follows, because s WAs = XA (t)−1 WA(t+·) . See for instance Abbondandolo and Majer (2003c, Proposition 1.2) for a complete proof (the case of a Hilbert space is treated in that reference, but the proof in the Banach setting presents no difference). Denote by C0k ([0, +∞[, E) the Banach space of all C k curves u: [0, +∞[ → E such that lim u(h) (t) = 0 ∀h ∈ {0, 1, . . . , k}. t→+∞


 PROPOSITION 1.6. Let A ∈ C 0 [0, +∞], L(E) be a path of bounded linear operators on the Banach space E such that L = A(+∞) is hyperbolic.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

5

(i) The bounded linear operator F A+ : C01 ([0, +∞[, E) → C00 ([0, +∞[, E),

u → u − Au,

is a left inverse. Moreover, F A+ admits a right inverse R+A such that 

WAs + R+A v(0) v ∈ C00 ([0, +∞[, E), v(0) = 0 = E. (ii) The evaluation map

ker F A+ → E,

(4)

u → u(0),

is a right inverse. Proof. We endow E with a Banach norm  ·  adapted to L. (i) Let us start by considering the case of the constant path A(t) ≡ L. By (2), the operator valued piecewise continuous function G: R → L(E),

G(t) = etL (R+ (t)Ps − R− (t)Pu ),

satisfies G(t) ≤ e−λ|t| , in particular it is integrable on R. Let v ∈ C00 ([0, +∞[, E). It is readily seen that the curve  +∞ u(t) = (G ∗ v)(t) = G(t − τ)v(τ) dτ 0

is continuously differentiable and solves the equation u (t) − Lu(t) = v(t).

(5)

u(t) ≤ GL1 (R,L(E)) v∞,[s,+∞[ + GL1 (]t−s,t[,L(E)) v∞ ,

(6)

Moreover, the inequality

shows that u ∈ C00 ([0, +∞[, E), so by (5), u ∈ C01 ([0, +∞[, E). We conclude that the operator R+L : C00 ([0, +∞[, E) → C01 ([0, +∞[, E),

v → G ∗ v,

is a right inverse of F L+ . Indeed, such a linear map is continuous by (5) and (6) with s = 0. Let us check that the operator v → R+L v(0) maps v ∈ Cc∞ (]0, +∞[, E) onto E u ; since E u is a direct complement of E s = WLs this implies that the right inverse R+L satisfies (4). Let ϕ be a smooth real function with supp ϕ ⊂ ]0, +∞[ so small that the operator  +∞ U := ϕ(τ)e−τL dτ ∈ L(E) 0

6

A. ABBONDANDOLO AND P. MAJER

is an isomorphism. The operator U preserves the splitting E = E u ⊕ E s . If ξ ∈ E u , setting v = −ϕU −1 ξ, there holds  +∞  +∞ R+L v(0) = e−τL Pu ϕ(τ)U −1 ξ dτ = ϕ(τ)e−τL dτ U −1 ξ = ξ, 0

0

proving the claim. Let us now consider the general case. Setting A s (t) = A(s + t), we have that lim F A+s = F L+

s→+∞

in the operator norm of L(C01 , C00 ). Since the set of left inverses is open, by our previous case we deduce that F A+s has a right inverse R+As for s large, such that R+As → R+L in the operator norm for s → +∞. Since the space of surjective operators is open, R+As satisfies (4) for s large. Fix such a large s. We can now define a right inverse R+A of F A+ by setting R+A v = u, where u is the solution of the linear Cauchy problem  u − Au = v, u(s) = R+As v s (0). The continuity of R+A is easily seen by the formula  t  + −1 + XA (τ)−1 v(τ) dτ . (RA v)(t) = XA (t) XA (s) RAs v s (0) + s

Finally, the fact that R+As satisfies (4) implies that also R+A satisfies (4). Indeed, let ξ ∈ E, and let v ∈ C00 ([0, +∞[, E) with v(0) = 0 be such that XA (s)ξ ∈ R+As v(0) + WAs s . Since v(0) = 0, the curve

(7)

 w(t) =

v(t − s) if t ≥ s, 0 if 0 ≤ t ≤ s,

belongs to C00 ([0, +∞[, E). Since w vanishes on [0, s], R+A w solves the equation u − Au = 0 on [0, s], so R+As v(0) = R+A w(s) = XA (s)R+A w(0). By (7) and (8), ξ ∈ R+A w(0) + XA (s)−1 WAs s = R+A w(0) + WAs , concluding the proof of (i).

(8)

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

7

(ii) The kernel of F A+ is ker F A+ = {XA (t)ξ | ξ ∈ WAs }, so if Q ∈ L(E) is a projector onto WAs , the linear map E → ker F A+ ,

ξ → XA (·)Qξ,

is a left inverse of the evaluation at 0, ker F A+ → E,

u → u(0).




REMARK 1.7. If P is a projector onto WAs , it can be shown that  +∞
 R+A v(t) = XA (t) R+ (t − τ)P − R− (t − τ)(I − P) XA (τ)−1 v(τ) dτ 0

defines a right inverse of F A+ . See Abbondandolo and Majer (2003c) for a more extensive discussion of the topics of this section. We conclude this section by establishing some properties of the operator d/dt − A(t) on the whole real line. 
PROPOSITION 1.8. Assume that A ∈ C 0 R, L(E) has hyperbolic asymptotic operators A(−∞) and A(+∞), both with finite-dimensional positive eigenspace. Then the bounded linear operator F A : C01 (R, E) → C00 (R, E),

u → u − Au,

is Fredholm of index

 ind F A = dim E u A(−∞) − dim E u A(+∞) . Moreover, WAu + WAs is closed and ker F A
WAu ∩ WAs , coker F A
E/(WAu + WAs ). (9)

 Proof. Since WAu
E u A(−∞) and WAs
E s A(+∞) , the first space is finitedimensional and the second one is finite-codimensional, with

 dim WAu = dim E u A(−∞) , codim WAs = dim E u A(+∞) . (10) Therefore, WAu + WAs is (closed and) finite-codimensional, and dim WAu ∩ WAs − codim(WAu + WAs ) = dim WAu − codim WAs .

(11)

8

A. ABBONDANDOLO AND P. MAJER

The kernel of F A is the linear subspace ker F A = {XA (t)ξ | ξ ∈ WAu ∩ WAs }, so dim ker F A = dim WAu ∩ WAs .

(12)

By Proposition 1.6(i), the operators F A+ : C01 ([0, +∞[, E) → C00 ([0, +∞[, E),

u → u − Au,

F A− : C01 (]−∞, 0], E) → C00 (]−∞, 0], E),

u → u − Au,

have right inverses R+A and R−A . If v is an element of C00 (R, E), any solution of u − Au = v has the form
 u(t) = XA (t) u(0) − R+A v(0) + R+A v(t), ∀t ≥ 0,
 u(t) = XA (t) u(0) − R−A v(0) + R−A v(t), ∀t ≤ 0. Such a curve u belongs to C01 (R, E) if and only if u(0) − R+A v(0) ∈ WAs and u(0) − R−A v(0) ∈ WAu . Therefore, v belongs to the range of F A if and only if the affine subspaces R+A v(0) + WAs and R−A v(0) + WAu have nonempty intersection, that is if and only if R+A v(0) − R−A v(0) belongs to WAs + WAu . So the range of F A is the linear subspace

 ran F A = v ∈ C00 (R, E) R+A v(0) − R−A v(0) ∈ WAu + WAs . Such a linear subspace is closed. By the second assertion in Proposition 1.6(i), the operator E C00 (R, E) → u , v → [R+A v(0) − R−A v(0)], WA + WAs is onto, so codim ran F A = codim(WAu + WAs ). All the statements follow from (10) – (13).

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1.3. MORSE VECTOR FIELDS

Let M be a Banach manifold of class C 2 , i.e., a paracompact Hausdorff topological space, locally homeomorphic to a Banach space E, endowed with an atlas whose transition maps are of class C 2 . See Lang (1999) for foundational results on Banach manifolds. A C 1 tangent vector field X on M defines a local flow φ solving
 ∂t φ(t, p) = X φ(t, p) , φ(0, p) = p, ∀p ∈ M, −∞ ≤ t− (p) < t < t+ (p) ≤ +∞,

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

9

where ]t− (p), t+ (p)[ denotes the maximal interval of existence of the above Cauchy problem. The functions t− and t+ are upper and lower semi-continuous, respectively. Denote by Ω(X) the subset of R × M which lies strictly between the graph of t− and the graph of t+ . Then Ω(X) is an open neighborhood of {0} × M, and the map φ: Ω(X) → M is of class C 1 . The vector field X is said complete (resp. positively complete, resp. negatively complete) if Ω(X) = R × M (resp. Ω(X) ⊃ [0, +∞[ × M, resp. Ω(X) ⊃ ]−∞, 0] × M). Let A be a positively invariant subset of M: this means that if p ∈ A then φ(t, p) ∈ A for every t ∈ [0, t+ (p)[. The vector field is said positively complete with respect to A if for every p ∈ M such that t+ (p) < +∞ there exists t ∈ [0, t+ (p)[ such that φ(t, p) ∈ A. Similarly, one defines a negatively complete vector field with respect to a negatively invariant subset. A rest point of X is a point x ∈ M such that X(x) = 0. The set of rest points of X will be denoted by rest(X). The Jacobian of X at a rest point x is the bounded linear operator on T x M defined by ∇X(x)ξ = [X, Y](x), where ξ ∈ T x M and Y is a tangent vector field on M such that Y(x) = ξ. Indeed, the fact that X(x) = 0 implies that this definition does not depend on the choice of extension Y of ξ. EXERCISE 1.9. Give an alternative definition of the Jacobian of a vector field at a rest point in terms of a local chart: if ϕ: U → E maps a neighborhood U of x ∈ rest(X) diffeomorphically onto an open subset of the Banach space E, define the operator ∇X(x) on T x M by
 ϕ∗ (∇X(x)ξ) = D(ϕ∗ X) ϕ(x) [ϕ∗ ξ],

∀ξ ∈ T x M,

where, for η ∈ T p M, ϕ∗ η is the vector in E defined by ϕ∗ η = Dϕ(p)[η]. Show that such a definition does not depend on the choice of the chart ϕ. A rest point x of X is said hyperbolic if the Jacobian of X at x is a hyperbolic operator. The corresponding splitting of the tangent space at x will be denoted by T x M = E ux ⊕ E sx . By the inverse mapping theorem, the hyperbolic rest points are isolated in rest(X). The Morse index m(x) ∈ N ∪ {+∞} of the hyperbolic rest point x is the dimension of the subspace E ux . The Morse co-index is the dimension of E sx . If all the rest points of X are hyperbolic, the vector field X is said a Morse vector field. 1.4. LOCAL DYNAMICS NEAR A HYPERBOLIC REST POINT

Let U be an open neighborhood of 0 in the Banach space E, and let X ∈ C 1 (U, E) be a vector field having 0 as a hyperbolic rest point. Denote by φ: Ω(X) → U the local flow of X. Let L := ∇X(0) = DX(0), with splitting E = E u ⊕ E s and projectors Pu , Ps , and let us endow E with an L-adapted norm  · . If V ⊂ E is a

10

A. ABBONDANDOLO AND P. MAJER

closed linear subspace, V(r) will denote the closed ball in V of radius r centered in 0, and ∂V(r) will be the relative boundary of V(r) in V. Consider the cones C u = {ξ ∈ E | Ps ξ ≤ Pu ξ} ⊃ E u ,

C s = {ξ ∈ E | Pu ξ ≤ Ps ξ} ⊃ E s .

We recall that if A ⊂ B ⊂ U the set A is said positively (negatively) invariant with respect to B if for every ξ ∈ A and for every t > 0, φ([0, t] × {ξ}) ⊂ B implies φ([0, t] × {ξ}) ⊂ A (resp. for every ξ ∈ A and for every t < 0, φ([t, 0] × {ξ}) ⊂ B implies φ([t, 0] × {ξ}) ⊂ A). LEMMA 1.10. For every r > 0 small enough there holds: (i) the set C u ∩ E(r) is positively invariant with respect to E(r); (ii) the set C s ∩ E(r) is negatively invariant with respect to E(r); (iii) if ξ belongs to the set C u ∩ ∂E(r) = ∂E u (r) × E s (r) then Pu φ(t, ξ) > r for every t ∈ ]0, 1], and Pu φ(t, ξ) < r for every t ∈ [−1, 0[; (iv) if ξ belongs to the set C s ∩ ∂E(r) = E u (r) × ∂E s (r) then Ps φ(t, ξ) < r for every t ∈ ]0, 1], and Ps φ(t, ξ) > r for every t ∈ [−1, 0[. Proof. Since t+ (0) = +∞ and t− (0) = −∞, we have t+ (ξ) > 1 and t− (ξ) < −1 for ξ small enough. A first order expansion of φ(t, ·) at 0 yields to φ(t, ξ) = etL ξ + o(ξ)t

for ξ → 0,

uniformly in t ∈ [−1, 1]. Therefore, if r > 0 is small enough, for every ξ ∈ C s ∩ E(r) and t ∈ [0, 1], (2) implies Ps φ(t, ξ) = Ps etL ξ + o(ξ)t = etL Ps ξ + o(Ps ξ)t ≤ e−λt Ps ξ + o(Ps ξ)t ≤ e−λt/2 Ps ξ, and similarly, for every ξ ∈ C u ∩ E(r) and t ∈ [0, 1], (3) implies Pu φ(t, ξ) ≥ eλt/2 Pu ξ. All the statements follow from the above inequalities and from the analogous inequalities holding for t ∈ [−1, 0].
REMARK 1.11. In the language of Conley theory, E(r) is an isolating neighborhood for the invariant set {0}, and ∂E u (r) × E s (r) is its exit set.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

11

1.5. LOCAL STABLE AND UNSTABLE MANIFOLDS

Given r > 0, the local unstable manifold and the local stable manifold of 0 are the sets

 u Wloc,r (0) = ξ ∈ E(r) t− (ξ) = −∞, φ(]−∞, 0] × {ξ}) ⊂ E(r), lim φ(t, ξ) = 0 , t→−∞

 s + Wloc,r (0) = ξ ∈ E(r) t (ξ) = +∞, φ([0, +∞[ × {ξ}) ⊂ E(r), lim φ(t, ξ) = 0 . t→+∞ When r is small, these sets are actually graphs of regular maps. THEOREM 1.12 (Local (un)stable manifold theorem). Assume that 0 is a hyperbolic rest point of the C k vector field X: U → E, k ≥ 1. For any r > 0 small s (0) is the graph of a C k map σs : E s (r) → E u (r) such that σs (0) = 0 enough, Wloc,r u (0) is the graph of a C k map σu : E u (r) → E s (r) and Dσs (0) = 0. Similarly, Wloc,r u u such that σ (0) = 0, Dσ (0) = 0. See Shub (1987, Chapter 5) for a proof based on the graph transform method. Here we will present a proof based on the study of the orbit space and on Proposition 1.6. Proof. We shall prove the conclusion for the local stable manifold, the case of the unstable one following by considering the vector field −X. The map Φ: C01 ([0, +∞[, U) → C00 ([0, +∞[, E),

u → u − X ◦ u,

is of class C k , and its differential at u ∈ C01 ([0, +∞[, U) is DΦ(u): C01 ([0, +∞[, E) → C00 ([0, +∞[, E), v → v − DX(u)v.
 Since DX u(t) converges to L = DX(0) for t → +∞, statement (i) of Proposition 1.6 implies that DΦ(u) is a left inverse, so Φ is a C k submersion. In particular, its set of zeros Φ−1 ({0}) is a C k submanifold of C01 ([0, +∞[, U). The set of zeros is nonempty, because it contains the curve 0. Actually T 0 Φ−1 ({0}) = ker DΦ(0) = ker(v → v − Lv) = {etL ξ | ξ ∈ E s }.

(14)

By statement (ii) of Proposition 1.6, the evaluation map ev0 : u → u(0) subordinates an immersion ev0 : Φ−1 ({0}) → U, which is injective by the uniqueness of the solution of Cauchy problems. Therefore, 
 W s (0) := ev0 Φ−1 ({0}) = ξ ∈ U t+ (ξ) = +∞, lim φ(t, ξ) = 0 t→+∞

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A. ABBONDANDOLO AND P. MAJER

is the image of an injective C k immersion. The point 0 belongs to W s (0), and by (14), 
T 0 W s (0) = D ev0 (0)T 0 Φ−1 ({0}) = ev0 T 0 Φ−1 ({0}) = E s . By the implicit function theorem, if r is small enough the path-connected component of W s (0) ∩ E(r) containing 0 — call it Zr — is the graph of a C k map σs : E s (r) → E u (r) such that σs (0) = 0 and Dσs (0) = 0. We claim that if r is so small that the conclusions of Lemma 1.10 hold, and s (0), which concludes that the Lipschitz norm of σs is less than 1, then Zr = Wloc,r s s (0) to the proof. Indeed, by definition Wloc,r (0) ⊂ Zr , a path connecting ξ ∈ Wloc,r s 0 within W (0) ∩ E(r) being provided by the orbit of ξ. On the other hand, notice that by definition Zr is positively invariant with respect to E(r). So if there exists s (0), by Lemma 1.10 there is some t > 0 for which φ(t, ξ) ∈
∂E u (r) × ξ ∈ Zr \ Wloc,r  E s (r) (the latter is the exit set of E(r)) and φ(t, ξ) ∈ Zr (Zr is positively invariant
 with respect to E(r)). Therefore Zr ∩ ∂E u (r) × E s (r) is nonempty, contradicting the fact that Zr is the graph of a map whose Lipschitz constant is less than 1, taking value 0 at 0.


1.6. THE GROBMAN – HARTMAN LINEARIZATION THEOREM

The Grobman – Hartman theorem says that up to a change of variables, the dynamics near a hyperbolic point is the dynamics given by a linear vector field. We will deduce this fact from the analogous statement for discrete dynamical systems. The proof is adapted from Shub (1987). Let us start with a result about the existence, uniqueness, and H¨older regularity of a semi-conjugacy between two perturbations of a linear operator. PROPOSITION 1.13. Let E = E u ⊕ E s be an invariant splitting for the bounded invertible operator T . Let Pu and Ps be the corresponding projectors, and assume that there exists µ < 1 such that max{Ps T Ps , Pu T −1 Pu } ≤ µ. Let ϕ and ψ be Lipschitz continuous maps from E to E such that: (i) ϕ − ψ∞ < +∞; (ii) lip ϕ < 1 − µ; (iii) lip ψ < 1/T −1 .

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13

Then there exists a unique bounded map g: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ). Moreover, g∞ ≤ and setting

(15)

ϕ − ψ∞ , 1 − (µ + lip ϕ)

 θ := max T  + lip ψ,

 T −1  , 1 − T −1  lip ψ

g is α-H¨older continuous for every α<

− log(µ + lip ϕ) . log θ

Notice that if E u  (0), then T  ≥ Pu T Pu  ≥ 1/µ, while if E s  (0), then T −1  ≥ Ps T −1 Ps  ≥ 1/µ. Therefore log θ ≥ − log µ, so the quantity − log(µ + lip ϕ)/ log θ appearing in the above proposition does not exceed 1. In general, the map g is not locally Lipschitz, even when ϕ and ψ are smooth. Proof. For an E-valued map f , we denote by fu and fs its components with respect to the splitting E = E u ⊕ E s , that is fu := Pu f , fs := Ps f . By applying the projectors Pu and Ps , (15) is equivalent to  (T u + ϕu ) ◦ (I + g) = (Pu + gu ) ◦ (T + ψ), (16) (T s + ϕs ) ◦ (I + g) = (Ps + gs ) ◦ (T + ψ). Since lip T −1 ψ ≤ T −1  lip ψ < 1, the map T + ψ = T (I + T −1 ψ) is a homeomorphism of E onto E. Actually, its inverse is Lipschitz continuous with 
lip(T + ψ)−1 = lip (I + T −1 ψ)−1 T −1 ≤

T −1  . 1 − T −1  lip ψ

(17)

By a simple algebraic manipulation, (16) is equivalent to the fixed point problem F(g) = g, where F(g)u = T u−1 (gu ◦ (T + ψ) − ϕu ◦ (I + g) + ψu ), F(g)s = (T s gs + ϕs ◦ (I + g) − ψs ) ◦ (T + ψ)−1 . Since F(g)u ∞ ≤ ≤ F(g)s ∞ ≤ ≤


µ (1 + lip ϕ)g∞ + ϕ − ψ∞ (µ + lip ϕ)g∞ + µϕ − ψ∞ , (T s  + lip ϕ)g∞ + ϕ − ψ∞ (µ + lip ϕ)g∞ + ϕ − ψ∞ ,

(18) (19)

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A. ABBONDANDOLO AND P. MAJER

F maps B(E, E), the Banach space of bounded maps from E to E, into itself. Actually, (18) and (19) imply that if g∞ ≤ R, with R :=

ϕ − ψ∞ , 1 − (µ + lip ϕ)

then F(g)∞ ≤ R. Moreover, the maps Fu = Pu F: B(E, E) → B(E, E u ) and Fs = Ps F: B(E, E) → B(E, E s ) are Lipschitz with lip Fu ≤ T u −1 (1 + lip ϕ) ≤ µ(1 + lip ϕ) < 1, lip Fs ≤ T s  + lip ϕ ≤ µ + lip ϕ < 1, so F: B(E, E) → B(E, E) is a contraction, proving that there exists a unique g ∈ B(E, E) satisfying (15). Since F maps the closed R-ball of C 0 ∩ B(E, E) into itself, the fixed point g is continuous and bounded by R. If h ∈ B(E, E) has modulus of continuity2 ω, then F(h)u has modulus of continuity
 t → µω (T  + lip ψ)t + µ lip ϕω(t) + µ(lip ϕ + lip ψ)t, (20) while by (17), F(h)s has modulus of continuity t → (µ + lip ϕ)ω(σt) + (lip ψ + lip ϕ)σt,

(21)

where σ := T −1 /(1−T −1  lip ψ). Comparing (20) and (21), we find that setting a := (lip ϕ + lip ψ)σ, the function t → (µ + lip ϕ)ω(θt) + at is a modulus of continuity for F(h). If moreover h∞ ≤ R, we have that F(h)∞ ≤ R, so F(h) has modulus of continuity t → min{(µ + lip ϕ)ω(θt) + at, 2R}. Therefore, if a modulus of continuity ω satisfies min{(µ + lip ϕ)ω(θt) + at, 2R} ≤ ω(t) ∀t ∈ [0, +∞[,

(22)

we deduce that the nonempty closed subset of B(E, E) {h ∈ B(E, E) | h∞ ≤ R, h has modulus of continuity ω} is F-invariant, hence the fixed point g has modulus of continuity ω. A function of the form ω(t) = ctα satisfies (22) if (µ + lip ϕ)θα < 1, 2

Here, moduli of continuity are always assumed to be nondecreasing.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

and c is large enough. The conclusion follows.

15


If we symmetrize the assumptions of the above proposition, a standard argument involving uniqueness yields to the following global version of the Grobman – Hartman theorem for discrete dynamical systems. COROLLARY 1.14. Let T be an invertible bounded operator on E satisfying the same assumptions of Proposition 1.13. Let ϕ and ψ be Lipschitz continuous maps from E to E such that (i) ϕ − ψ∞ < +∞; (ii) lip ϕ < min{1 − µ, 1/T −1 }, lip ψ < min{1 − µ, 1/T −1 }. Then there exists a unique bounded map g: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ). Moreover, g is H¨older continuous, and I + g is homeomorphism of E onto E. Proof. Applying Proposition 1.13 to the pair (ϕ, ψ) and to the pair (ψ, ϕ), we find H¨older continuous bounded maps g: E → E and h: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ), (T + ψ) ◦ (I + h) = (I + h) ◦ (T + ϕ). It follows that (I + g) ◦ (I + h), which is of the form I + k with k ∈ B(E, E), satisfies (T + ϕ) ◦ (I + k) = (I + k) ◦ (T + ϕ). By the uniqueness statement of Proposition 1.13 applied to the pair (ϕ, ϕ), k must be the zero map, that is (I + g) ◦ (I + h) = I. Similarly, (I + h) ◦ (I + g) = I, so I + g is a homeomorphism of E onto E with inverse I + h.
Now we can derive a global version of the Grobman – Hartman theorem for flows. THEOREM 1.15. Let L be a hyperbolic operator on E. Let  ·  be an L-adapted norm on E, satisfying (1) and (2) for some positive λ. Let B1 : E → E and B2 : E → E be Lipschitz continuous maps such that (i) B1 − B2 ∞ < +∞;

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(ii) lip B1 < λ, lip B2 < λ. Then the flows φ1 , φ2 : R × E → E of the vector fields X1 (ξ) = Lξ + B1 (ξ) and X2 (ξ) = Lξ + B2 (ξ) are conjugated. More precisely, there is a unique bounded map g: E → E such that

 φ1 t, (I + g)(ξ) = (I + g) φ2 (t, ξ ∀(t, ξ) ∈ R × E, and I + g is a homeomorphism of E onto E. Moreover, g is α-H¨older continuous for every λ − lip B1 . α< L + lip B2 Proof. A c-Lipschitz vector field X produces a globally defined flow φ, with lip φ(t, ·) ≤ ec|t| . If two c-Lipschitz vector fields X1 , X2 have bounded distance, then φ1 (t, ·) − φ2 (t, ·)∞ ≤ X1 − X2 ∞ |t|ec|t| . Let ψi (t, ξ) = φi (t, ξ) − etL ξ, for i = 1, 2. By our initial considerations, for every t the maps ψ1 (t, ·) and ψ2 (t, ·) are Lipschitz and have bounded distance. The maps ψi satisfy  t e(t−s)L Bi (e sL ξ + ψi (s, ξ)) ds ∀(t, ξ) ∈ R × E. (23) ψi (t, ξ) = 0

By (23),  t
 e(t−s)L  e sL  + lip ψi (s, ·) ds lip ψi (t, ·) ≤ lip Bi 0  ≤ |t| lip Bi sup e sL  sup e sL  + sup lip ψi (s, ·) |s|≤|t|

|s|≤|t|

|s|≤|t|


 = |t| lip Bi 1 + sup lip ψi (s, ·) 1 + o(1) , 

|s|≤|t|

for t → 0. Taking the supremum for all |t| ≤ τ, we obtain
 sup lip ψi (t, ·) ≤ lip Bi |τ| 1 + o(1) for τ → 0. |t|≤τ

(24)

Since lip Bi < λ, the last inequality implies that there exists τ > 0 such that lip ψi (t, ·) < 1 − e−λ|t| , lip ψi (t, ·) < 1/e−tL ,

∀ 0 < |t| ≤ τ, i = 1, 2.

By Corollary 1.14 applied to T = etL , µ = e−λ|t| , ϕ = ψ1 (t, ·), and ψ = ψ2 (t, ·), for every 0 < |t| ≤ τ there exists a unique gt ∈ B(E, E) such that

 (25) φ1 t, (I + gt )(ξ) = (I + gt ) φ2 (t, ξ) ,

17

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and I +gt is a homeomorphism of E onto E. If n ∈ Z\{0} and |nt| ≤ τ, (25) implies 

φ1 nt, (I + gt )(·) = (I + gt ) φ2 (nt, ·) , so by uniqueness gt = gnt . If p, q are rational numbers in [−τ, τ] \ {0}, they have a common sub-multiple, so g p = gq . Therefore g p = g for every rational p ∈ [−τ, τ] \ {0}. By the continuity of φ1 and φ2 with respect of t, 

φ1 t, (I + g)(·) = (I + g) φ2 (t, ·) holds for every |t| ≤ τ, hence by taking iterates, for every t ∈ R. There remains to estimate the H¨older exponent of g. Let 0 < t ≤ τ. By Proposition 1.13 gt is α-H¨older for every
 − log e−λt + lip ψ1 (t, ·) α< , log θ(t) where

 θ(t) = max etL  + lip ψ2 (t, ·),

 e−tL  . 1 − e−tL  lip ψ2 (t, ·)

Since gt = g, g is α-H¨older for every α < β := lim sup t→0+

− log(e−λt + lip ψ1 (t, ·)) . log θ(t)

By (24),

 − log e−λt + lip ψ1 (t, ·) ≥ − log e−λt + lip B1 t + o(t) = (λ − lip B1 )t + o(t) (26) for t → 0+ . Since

e±tL  ≤ etL = 1 + Lt + o(t),

for t → 0+ , by (24) there holds etL  + lip ψ2 (t, ·) ≤ 1 + (L + lip B2 )t + o(t), 1 + Lt + o(t) e−tL  ≤ = 1 + (L + lip B2 )t + o(t), −tL 1 − lip B2 t + o(t) 1 − e  lip ψ2 (t, ·) for t → 0+ . Therefore


log θ(t) ≤ log 1 + (L + lip B2 )t + o(t) = (L + lip B2 )t + o(t),

(27)

for t → 0+ . The inequalities (26) and (27) imply that
 − log e−λt + lip ψ1 (t, ·) λ − lip B1 (λ − lip B1 )t + o(t) ≥ lim+ = β = lim sup , t→0 + log θ(t) (L + lip B )t + o(t) L + lip B2 2 t→0 concluding the proof. It is then straightforward to deduce the following local linearization result.




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COROLLARY 1.16. Assume that 0 is a hyperbolic rest point of the C 1 vector field X: U → E, and let L = DX(0). If r > 0 is small enough, the local flow φ restricted to E(r) is conjugated to the linear flow (t, ξ) → etL ξ by a bi-H¨older continuous homeomorphism. More precisely, there exists a bi-H¨older continuous
 homeomorphism h: E(r) → h E(r) ⊂ E such that
 h φ(t, ξ) = etL h(ξ) ∀(t, ξ) ∈ Ω(X|E(r) ). We conclude the discussion about the local dynamics at a rest point with the following proposition. PROPOSITION 1.17. For every r > 0 small enough there holds: for every sequence (ξn ) ⊂ E converging to 0 and for every sequence (tn ) ⊂ [0, +∞[ such that φ([0, tn ] × {ξn }) ⊂ E(r) and φ(tn , ξn ) ∈ ∂E(r), there holds
 u dist φ(tn , ξn ), Wloc,r (0) ∩ ∂E(r) → 0. Proof. If the vector field is linear, X(ξ) = Lξ, the conclusion is immediate: u (0) = E u (r), and for any (ξ ) ⊂ E converging to 0 and indeed in this case Wloc,r n any (tn ) ⊂ [0, +∞[, by (2) we have lim sup dist(etn L ξn , E u ) = lim sup Ps etn L ξn  ≤ lim sup e−tn λ Ps ξn  = 0. n→∞

n→∞

n→∞

By the Grobman – Hartman theorem, if r0 > 0 is small enough the local flow φ restricted to E(r0 ) is conjugated to its linearization (t, ξ) → etL ξ, by a biuniformly continuous homeomorphism. By the local (un)stable manifold theorem, u we may also assume that r0 is so small that Wloc,r (0) is the graph of a uniformly 0 u u s continuous map σ : E (r0 ) → E (r0 ). Let r < r0 and set ηn := φ(tn , ξn ) ∈ ∂E(r), with ξn → 0 and tn ≥ 0. By Lemma 1.10, Pu ηn  = r. By the linear case and by the uniform continuity of the u conjugacy, there exists (η n ) ⊂ Wloc,r (0) such that ηn −η n  is infinitesimal. Setting 0
 u u u u u η n = P ηn , σ (P ηn ) ∈ Wloc,r (0) ∩ ∂E(r), by the uniform continuity of σ we have 
u (0) ∩ ∂E(r) dist ηn , Wloc,r ≤ ηn − η n s s ≤ ηn − η n  + Pu η n − Pu η n  + P ηn − P ηn  = ηn − η n  + Pu η n − Pu ηn  + σu (Pu η n ) − σu (Pu ηn ) → 0, concluding the proof.




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19

1.7. GLOBAL STABLE AND UNSTABLE MANIFOLDS

Let us assume that X is a C 1 tangent vector field on the Banach manifold M, and that x is a hyperbolic rest point of X. We shall identify a neighborhood of x with a neighborhood of 0 in the Banach space E x = T x M, identifying x with 0. We shall consider a ∇X(x)-adapted norm on E x , and we will use the notation introduced in the above sections: for instance, E x (r) ⊂ M will denote the closed r ball centered in x. The unstable and the stable manifolds of the rest point x are the subsets of M

 W u (x) = p ∈ M t− (p) = −∞ and lim φ(t, p) = x , t→−∞

 s + W (x) = p ∈ M t (p) = +∞ and lim φ(t, p) = x . t→+∞

THEOREM 1.18. Let x ∈ M be a hyperbolic rest point of the C k vector field X, k ≥ 1, on the Banach manifold M. Then W u (x) and W s (x) are the images of injective C k immersions of manifolds which are homeomorphic to E ux and E sx , respectively. Proof. By Theorem 1.12, if r is small enough the local unstable manifold u (x) is the graph of a C k map σu : E u (r) → E s (r). Since Wloc,r 

u (x), 0 ≤ t < t+ (p) , W u (x) = φ(t, p) p ∈ Wloc,r u (x) by the the set W u (x) inherits the structure of a C k manifold from that of Wloc,r maps {φ(t, ·)}, and the inclusion of W u (x) into M is a C k injective immersion. u (x) is the C k diffeomorphism θ(ξ) = ξ + σu (ξ), the map If θ: E u (r) → Wloc,r




A := ξ ∈ E ux log ξ < t+ θ(rξ/ξ) → W u (x),


 ξ → φ log ξ, θ(rξ/ξ) ,

is a homeomorphism from a star-shaped open subset of E ux - thus homeomorphic to E ux itself - onto W u (x). The analogous results for W s (x) follow by considering the vector field −X.
REMARK 1.19. If M is a Hilbert manifold, then the regularity of the norm implies that W u (x) and W s (x) are actually images of C k immersions of E ux and E sx , respectively. In general W u (x) and W s (x) need not be embedded submanifolds: actually, they need not be locally closed. A Lyapunov function for X is a C 1 function f : M → R such that D f (p)[X(p)] < 0 for every p ∈ M \ rest(X). In this case, of course crit( f ) ⊂ rest(X). If X is

20

A. ABBONDANDOLO AND P. MAJER

a Morse vector field, the two sets actually coincide. Indeed, we can assume that M = E is a Banach space, so if x ∈ rest(X) and t ∈ R we have D f (x+tv)[X(x+tv)] = D f (x)[X(x+tv)]+o(t) = tD f (x)[DX(x)v]+o(t) for t → 0. The principal part of the right-hand side is an odd function of t. Since the above quantity has to be negative for every t  0, such a principal part has to be identically zero. Since DX(x) is an isomorphism, we deduce that D f (x) = 0. THEOREM 1.20. Assume that X admits a Lyapunov function f , and that for every r0 > 0 small enough there holds 



u s p ∈ Wloc,r sup f (p) p ∈ Wloc,r (x) ∩ ∂E (r ) < inf f (p) (x) ∩ ∂E x (r0 ) . (28) x 0 0 0 Then if r > 0 is small enough: (i) for every p ∈ M, the closed set I = {t ∈ ]t− (p), t+ (p)[ | φ(t, p) ∈ E x (r)} is an interval, and its interior is {t ∈ ]t− (p), t+ (p)[ | φ(t, p) ∈ E˚ x (r)}; (ii) if I is upper bounded, then φ(max I, p) ∈ ∂E ux (r) ∩ E s (r); conversely, if φ(t, p) ∈ ∂E ux (r) ∩ E s (r), then t = max I; (iii) if I is lower bounded, then φ(min I, p) ∈ E ux (r) ∩ ∂E s (r); conversely, if φ(t, p) ∈ E ux (r) ∩ ∂E s (r), then t = min I; u (x), and W s (x) ∩ E (r) = W s (x); (iv) W u (x) ∩ E x (r) = Wloc,r x loc,r

(v) W u (x) and W s (x) are submanifolds of M. Proof. Let r0 be so small that (28) and the conclusions of Lemma 1.10, Theorem 1.12, and Proposition 1.17 hold. Since f is of class C 1 , up to choosing a smaller r0 we may also assume that f is uniformly continuous on E x (r0 ). We shall prove the first assertion in (i) arguing by contradiction. In fact, assume that there exist an infinitesimal sequence of positive numbers (ρn ), ρn < r0 , a sequence of points (ξn ) ∈ ∂E x (ρn ), and a sequence of positive numbers (tn ) such that φ(tn , ξn ) ∈ ∂E x (ρn ) and φ(]0, tn [ × {ξn }) ∩ E x (ρn ) = ∅. Lemma 1.10(iv) implies that (at least for n large) ξn ∈ ∂E ux (ρn ) × E sx (ρn ) ⊂ C u ∩ E x (r0 ). By Lemma 1.10(i), C u ∩ E x (r0 ) is positively invariant with respect to E x (r0 ), and by (iii) if t > 0 and φ([0, t] × {ξn }) ⊂ C u ∩ E x (r0 ) then φ(t, ξn )  E x (ρn ). Therefore, there exists an ∈ ]0, tn [ such that φ([0, an ] × {ξn }) ⊂ E x (r0 ) and φ(an , ξn ) ∈ ∂E x (r0 ). Similarly, there exists bn ∈ [an , tn [ such that φ(bn , ξn ) ∈ ∂E x (r0 ) and φ([b, tn ] × {ξn }) ⊂ E x (r0 ). Since ξn → 0 and φ(tn , ξn ) → 0, Proposition 1.17 implies that

 u s dist φ(an , ξn ), Wloc,r (x)∩∂E x (r0 ) → 0, dist φ(bn , ξn ), Wloc,r (x)∩∂E x (r0 ) → 0. 0 0 Therefore, by (28), taking into account the fact that f is uniformly continuous on E x (r0 ), lim sup f (φ(an , ξn )) ≤ n→∞

sup u Wloc,r (x)∩∂E x (r0 ) 0

f <

inf

s Wloc,r (x)∩∂E x (r0 ) 0

f ≤ lim inf f (φ(bn , ξn )), n→∞

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

21



 a contradiction because an ≤ bn implies f φ(an , ξn ) ≥ f φ(bn , ξn ) . The second statement in (i), and statements (ii), (iii) are immediate consequences of Lemma 1.10. The inclusions u Wloc,r (x) ⊂ W u (x) ∩ E x (r),

s Wloc,r (x) ⊂ W s (x) ∩ E x (r),

are obvious. The opposite inclusions follow from statement (i). Then W u (x) and W s (x) are submanifold of M, because W u (x) = {φ(t, p) | p ∈ W u (x) ∩ E(r), 0 ≤ t < t+ (p)}, W s (x) = {φ(t, p) | p ∈ W s (x) ∩ E(r), t− (p) < t ≤ 0}, and because φ(t, ·) is a diffeomorphism.




REMARK 1.21. The weak inequality always holds in (28). The strict inequality holds if either: (i) x has finite Morse index; (ii) M is a Hilbert manifold, f is twice differentiable at x and the second differential of f at x satisfies D2 f (x)[ξ, ξ] ≤ −λξ2 for every ξ ∈ E ux , for some positive constant λ. u (x) is a compact set, so Indeed, in the first case Wloc,r 0

sup u Wloc,r (x)∩∂E x (r0 ) 0

f =

max

u Wloc,r (x)∩∂E x (r0 ) 0

f < f (x) ≤

inf

s Wloc,r (x)∩∂E x (r0 )

f.

0

In the second case, a second order expansion of f at x yields to the same conclusion.

2. The Morse complex in the case of finite Morse indices 2.1. THE PALAIS – SMALE CONDITION

Assume that f is a Lyapunov function for the vector field X on the Banach manifold M. A Palais – Smale (PS) sequence at level c is a sequence (pn ) ⊂ M such that ( f (pn )) converges to c and (D f (pn )[X(pn )]) is infinitesimal. We shall say that (X, f ) satisfies the Palais – Smale (PS) condition at level c if every Palais – Smale sequence at level c is compact. If (X, f ) satisfies the (PS) condition at every c ∈ [a, b], then rest(X)∩ f −1 ([a, b]) is compact, so if X is also Morse this set is finite.

22

A. ABBONDANDOLO AND P. MAJER

REMARK 2.1. Assume that Jn = φ([0, tn ] × {pn }), tn > 0, is contained in a strip {a ≤ f ≤ b}, and that lim

n→∞

f (pn ) − f (φ(tn , pn )) = 0. tn

(29)

Then there is a (PS) sequence qn ∈ Jn . Indeed, by the mean value theorem there is sn ∈]0, tn [ such that


 f (pn ) − f φ(tn , pn ) D f φ(sn , pn ) X φ(sn , pn ) = , tn and by (29), qn = φ(sn , pn ) is a (PS) sequence. Actually, the above observation could be used to give a weaker formulation of the (PS) condition, which does not require f to be differentiable, and could be used to study flows in the continuous category. 2.2. THE MORSE – SMALE CONDITION

We recall that two closed linear subspaces V1 , V2 of a Banach space E are said transverse if V1 +V2 = E and V1 ∩V2 is complemented in E. Two C 1 submanifolds M1 and M2 of the Banach manifold M are said transverse if for every p ∈ M1 ∩ M2 the closed linear subspaces T p M1 and T p M2 are transverse in T p M. Let X be a Morse vector field having only rest points with finite Morse index and admitting a Lyapunov function. We will say that X satisfies the Morse – Smale condition up to order k ∈ N if for every pair of rest points x, y satisfying m(x) − m(y) ≤ k, the submanifolds W u (x) and W s (y) are transverse. In this case, the implicit function theorem implies that W u (x) ∩ W s (y) — if nonempty — is a submanifold of dimension m(x) − m(y). Notice that the presence of a Lyapunov function implies that W u (x) ∩ W s (x) = {x}, and such an intersection is always transverse. Notice also that the fact that φ(t, ·) is a diffeomorphism implies that if W u (x) ∩ W s (y) meet transversally at some p ∈ M, they meet transversally at every point of the orbit of p. 2.3. THE ASSUMPTIONS

 and let X  be a C 1 vector field Let M be an open subset of the Banach manifold M,    on M (possibly, M = M). Denote by A the open subset M \ M, and denote by X  to M. the restriction of X We shall construct the Morse complex for X on M under the following assumptions:  and X  is positively (A1) A is positively invariant with respect to the flow of X, complete with respect to A;

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

23

(A2) X is a Morse vector field on M; (A3) every rest point of X has finite Morse index; (A4) X admits a Lyapunov function f ∈ C 1 (M) ∩ C 0 (M); (A5) f is bounded below on M; (A6) (X, f ) satisfies the (PS) condition at every level c ∈ f (M); (A7) X satisfies the Morse – Smale condition up to order 0.  will be denoted by φ. By (A1), the local flow of X is just The local flow of X the restriction of φ to   p ∈ M, φ(t, p) ∈ M . Ω(X) = (t, p) ∈ Ω(X)  and A is a sublevel In most applications f is actually defined on the whole M of f . Notice that (A6) and the fact that f ∈ C 0 (M) imply that there are no rest points on the boundary of M: such a rest point would be the limit of a (PS) sequence in M, which does not converge in M. Notice also that (A7) means asking that W u (x) does not meet W s (y) whenever x  y are rest points with m(x) ≤ m(y). 2.4. FORWARD COMPACTNESS

The (PS) condition plays a crucial role in the following compactness result. PROPOSITION 2.2. Assume (A1) – (A7). Then (i) for every p ∈ M, φ(t, p) either converges to a rest point of X for t → +∞ or eventually enters A;  (tn ) ⊂ [0, +∞[, and
φ(tn , pn ) ⊂ M, then (ii) if (pn ) ⊂ M converges to p ∈ M,
  the sequence φ(tn , pn ) is compact in M. Proof. (i) Let p ∈ M. Assume that φ(t, p) never enters A: by (A1) this implies that t+ (p) = +∞. By Remark 2.1, with pn = p, tn → +∞, a = inf f , b = f (p), and by (PS) we can find a sequence sn → +∞ such that φ(sn , p) converges to a rest
 point x ∈ M. The function t → f φ(t, p) converges for t → +∞, being monotone, therefore

 lim f φ(t, p) = lim f φ(sn , p) = f (x). t→+∞

n→∞

Assume by contradiction that φ(t, p) does not converge to x for t → +∞. Then we can find r > 0 (as small as we like), two sequences an ≤ bn ≤ an+1 , an → +∞, such that φ(an , p) ∈ ∂E x (r), φ(bn , p) ∈ ∂E x (2r), φ([an , bn ] × {p}) ⊂ E x (2r) \ E x (r).

24

A. ABBONDANDOLO AND P. MAJER

Choosing r so small that X is bounded on E x (2r) ⊂ M, one has that bn − an is bounded away from 0. Since


 lim f φ(an , p) = lim f φ(bn , p) = lim f φ(t, p) = f (x), n→∞

n→∞

we have lim

n→∞

t→+∞



 f φ(bn , p) − f φ(an , p) = 0, bn − an

so by Remark 2.1 there is a (PS) sequence in E x (2r) \ E x (r), converging by (PS) to a rest point. Since r is arbitrarily small, x is not isolated in rest(X), contradicting (A2). (ii) If (tn ) is bounded, then lim sup tn < t+ (p).

(30)

n→∞

Indeed, if by contradiction t+ (p) ≤ lim supn→∞ tn , t+ (p) is finite, so by (A1) there exists s ∈ [0, t+ (p)[ such that φ(s, p) ∈ A. Then φ(s, pn ) eventually belongs to A, so s > tn for n large, and lim supn→∞ tn ≤ s < t+ (p), a contradiction.
 When (tn ) is bounded, the continuity of φ and (30) imply that φ(tn , pn ) is  so we may assume that tn → +∞. compact in M, By Remark 2.1 and (PS) there exists a sequence an ∈ [0, tn ] such that, up to a subsequence, φ(an , pn ) converges to a rest point x ∈ M, with inf f ≤ f (x) ≤ f (p). Since there are finitely many rest points in this strip, we may assume that f (x) is minimal, that is for no sequence a n ∈ [0, tn ], φ(a n , pn ) has a subsequence converging to a rest point y ∈ M with f (y) < f (x). If φ(tn , pn ) converges to x then there is nothing to prove, otherwise up to a subsequence we can find r > 0 (as small as we like) and bn ∈ [an , tn ] such that φ(bn , pn ) ∈ ∂E x (r) and φ([an , bn ] × {pn }) ⊂ E x (r). By Proposition 1.17,
 the sequence φ(bn , pn ) is compact, since its distance from the compact set u (x) ∩ ∂E (r) tends to 0 (here we are using the fact that x has finite Morse Wloc,r x
 index). So a subsequence of φ(bn , pn ) converges to a point q with f (q) ≤

max

u (x)∩∂E (r) Wloc,r x

f < f (x).

The sequence tn − bn is bounded: otherwise by Remark 2.1 and (PS) there would
 exist cn ∈ [bn , tn ] such that a subsequence of φ(cn , pn ) converges to a rest point y with f (y) ≤ f (q) < f (x), contradicting the minimality of f (x). Therefore
  φ(tn , pn ) = φ tn − bn , φ(bn , pn ) is compact in M.
The above result has the following immediate consequence.  COROLLARY 2.3. For every x ∈ rest(X), W u (x) ∩ M has compact closure in M.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

25

Another consequence is the following convergence result for forward orbits: if (pn ) ⊂ M converges to p ∈ M, up to a subsequence the forward orbit of pn converges to a “broken orbit” consisting of h + 1 flow lines, h ≥ 0, matching at h rest points xh , . . . , x1 . The first of these flow lines is the forward orbit of p, the last one either converges to a rest point x0 , which is also the common limit of φ(t, pn ) for t → +∞, or eventually enters A, together with all the orbits of pn . More precisely, the situation is described by the following corollary. COROLLARY 2.4. Assume that (pn ) ⊂ M converges to some p ∈ M. Then there exists a subsequence (pkn ) such that one of the following two alternatives holds: (a) t+ (pkn ) = +∞, and there exists x0 ∈ rest(X) such that φ(t, pkn ) converges to x0 for t → +∞, for every n ∈ N; (b) for every n ∈ N, φ(t, pkn ) eventually enters A. Moreover, there exist h ∈ N, a set {x j }1≤ j≤h ⊂ rest(X), with f (x1 ) < · · · < f (xh ), sequences of real numbers t0n > t1n > · · · > thn = 0, and points q0 , q1 , . . . , qh = p in M such that: (i) q j ∈ W s (x j ) ∩ W u (x j+1 ) for every 1 ≤ j ≤ h − 1; (ii) qh = p ∈ W s (xh ), unless case (b) holds and h = 0, in which case φ(t, qh ) = φ(t, p) eventually enters A; (iii) q0 ∈ W u (x1 ) if h ≥ 1; in case (a) q0 ∈ W s (x0 ), in case (b) φ(t, q0 ) eventually enters A; j

(iv) limn→∞ φ(tn , pkn ) = q j for every 0 ≤ j ≤ h. The proof is an easy application of Proposition 2.2, together with an induction argument. Details are left to the reader. 2.5. CONSEQUENCES OF COMPACTNESS AND TRANSVERSALITY

 we will denote by φ([0, +∞[ × B) its forward evolution, Given a subset B ⊂ M, although this set should more properly be indicated by
 . φ ([0, +∞[ × B) ∩ Ω(X) The Morse – Smale condition up to order zero, assumption (A7), has the following consequence. LEMMA 2.5. Assume (A1) – (A7). Let x, y be distinct rest points of X, with m(x) ≤ m(y). Then there exists r > 0 such that
 φ [0, +∞[ × E x (r) ∩ Ey (r) = ∅.

26

A. ABBONDANDOLO AND P. MAJER

Proof. Assume the contrary: there exist a sequence (pn ) ⊂ M converging to
 x and a sequence (tn ) ⊂ [0, +∞[ such that φ(tn , pn ) converges to y. By Corollary 2.4, a subsequence of the sequence of forward orbits of pn converges to a “broken orbit” passing through x and y. In particular, there are pairwise distinct rest points z1 = x, z2 , . . . , zk = y, k ≥ 2, such that W u (zi ) ∩ W s (zi+1 )  ∅ for every 1 ≤ i ≤ k − 1. The Morse – Smale condition up to order zero implies that m(x) = m(z1 ) > · · · > m(zk ) = m(y), a contradiction.
In particular, the closure of the unstable manifold of a rest point x of index k does not contain rest points of index greater than or equal to k, other than x itself. Let us state a stronger assumption, which will be later removed: (A8) every rest point y does not belong to the closure of the union of the unstable manifolds of rest points x  y with m(x) ≤ m(y):  y W u (x). x ∈ rest(X) \ {y} m(x) ≤ m(y)

Since the closure of a finite union is the union of the closures, by Lemma 2.5 condition (A8) is implied by the Morse – Smale condition up to order zero (A7) when X has finitely many rest points of index k, for every k ∈ N. In general it is strictly more restrictive. Assumption (A8) implies the following result. PROPOSITION 2.6. Assume (A1) – (A8). Then there exists a positive function ρ: rest(X) → ]0, +∞[ such that 

 φ [0, +∞[ × E x ρ(x) ∩ Ey ρ(y) = ∅ for all pairs of rest points x  y with m(x) ≤ m(y). Proof. By (A8) there exists a function σ: rest(X) → ]0, +∞[ such that 
 W u (x) = ∅ ∀y ∈ rest(X). Ey σ(y) ∩

(31)

x ∈ rest(X) \ {y} m(x) ≤ m(y)

Let us prove that for every x ∈ rest(X) there is a positive number θ(x) such that 

 φ [0, +∞[ × E x θ(x) ∩ Ey σ(y) = ∅ ∀y ∈ rest(X) \ {x}, m(y) ≥ m(x). (32) Then the function ρ(x): rest(X) → ]0, +∞[, x → min{σ(x), θ(x)}, will satisfy the requirements. We argue by contradiction, assuming that there exists x ∈ rest(X) for which (32) does not hold, no matter how small θ(x) is. Since there are finitely many rest

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

27

points in {p ∈ M | f (p) ≤ f (x)}, we can find a sequence (pn ) ⊂ M converging to x and a sequence (tn ) ⊂ [0, +∞[ such that φ(tn , pn ) ∈ Ey (σ(y)), for some y ∈ rest(X) \ {x} with m(y) ≥ m(x). By Corollary 2.4, a subsequence of the sequence of the forward orbits of pn converges to a “broken orbit” starting from x and passing
 through Ey σ(y) . In particular, there are rest points z1 = x, . . . , zk  y, k ≥ 1, such that W u (zi ) ∩ W s (zi+1 )  ∅ for 1 ≤ i ≤ k − 1, and
 W u (zk ) ∩ Ey σ(y)  ∅. (33) By the Morse – Smale condition up to order 0, m(zk ) ≤ m(x) ≤ m(y), and since zk  y, (33) contradicts (31).


2.6. CELLULAR FILTRATIONS

Cellular filtrations are a useful tool to compute the singular homology of a topological space. See Dold (1980, Section V.1) for a more extensive discussion and for the proof of the results stated in this section. Let T be a topological space. A sequence F = {Fn }n∈Z of subsets of T is said a cellular filtration of T if: (i) Fn ⊂ Fn+1 for every n ∈ Z; (ii) every singular simplex in T is a simplex in Fn for some n; (iii) the k-th singular homology group Hk (Fn , Fn−1 ) vanishes for every k  n. Notice that (ii) is fulfilled when T is the union of the family {Fn } and each Fn is open. The space F−1 may be empty. The spaces Fn for n ≤ −2 will be actually irrelevant in the construction. Singular homology is always meant to have integer coefficients. If F = {Fn }n∈Z is a cellular filtration of T , we denote by Wk F the Abelian group Wk F := Hk (Fk , Fk−1 ). The homomorphism ∂k : Wk F → Wk−1 F is given by the composition Hk (Fk , Fk−1 ) → Hk−1 (Fk−1 ) → Hk−1 (Fk−1 , Fk−2 ), where the first map is the boundary homomorphism of the pair (Fk , Fk−1 ), and the second map is induced by the inclusion. It is readily seen that ∂k ∂k+1 = 0, so W∗ F is a chain complex of Abelian groups, said the cellular complex of the filtration F. A cellular map g: (T, F ) → (T , F ) is a continuous map from T to T mapping each Fn into Fn . Such a map induces homomorphisms Wk g: Wk F → Wk F ,

Wk g = g∗ : Hk (Fk , Fk−1 ) → Hk (Fk , Fk−1 ),

28

A. ABBONDANDOLO AND P. MAJER

which are readily seen to form a chain map W∗ g: W∗ F → W∗ F . This makes W a functor from the category of cellular filtrations and cellular maps to the category of chain complexes of Abelian groups and chain maps. THEOREM 2.7. If F = {Fn }n∈Z is a cellular filtration of the topological space T , then there is an isomorphism Hk ({W∗ F , ∂∗ })
Hk (T, F−1 ). Such isomorphisms form a natural transformation between the functor HW and the singular homology functor H, in the sense that if g: (T, F ) → (T , F ) is a cellular map, then the diagram Hk ({W∗ F , ∂∗ }) Hk Wk g



Hk ({W∗ F , ∂∗ })







/ Hk (T, F−1 ) 

g∗

/ Hk (T , F ) −1

commutes. A cellular homotopy h between two cellular maps g0 , g1 : (T, F ) → (T , F ) is cellular map h: ([0, 1] × T, F) → (T , F ), F being the cellular filtration {[0, 1] × Fn }n∈Z , such that h(0, ·) = g0 and h(1, ·) = g1 . If there is a cellular homotopy between g and g , the homotopy invariance of singular homology implies that W∗ g = W∗ g . A cellular map g: (T, F ) → (T , F ) is said a cellular homotopy equivalence if there are a cellular map g : (T , F ) → (T, F ), said a cellular homotopy inverse of g, and cellular homotopies h between g ◦ g and id(T,F ) and h between g ◦ g and id(T ,F ) . By functoriality and homotopy invariance, if g is a cellular homotopy equivalence then W∗ g is an isomorphism. 2.7. THE MORSE COMPLEX

Denote by restk (X) the set of rest points of X of Morse index k, and let Ck (X) be the free Abelian group generated by the elements of restk (X). Let ρ: rest(X) → ]0, +∞[ be a function satisfying 

 φ [0, +∞[ × E x ρ(x) ∩ Ey ρ(y) = ∅, ∀x  y ∈ rest(X), m(x) ≤ m(y), (34)  whose existence is established by Proposition 2.6. Consider the subsets of M  Mk = Mk (ρ) := A ∪ φ([0, +∞[ × E˚ x (ρ(x))) ∀k ≥ 0, Mk = A ∀k < 0, x ∈ rest(X) m(x) ≤ k

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

29

 and M∞ = M∞ (ρ) := k∈Z Mk . Each Mk is open and positively invariant. We shall denote by Dk the closed unit ball of Rk , and by ωk the generator of Hk (Dk , ∂Dk ) corresponding to the standard orientation of Rk . Here is the main result of this second part. THEOREM 2.8. Assume (A1) – (A8). Let ρ: rest(X) →]0, +∞[ be a function satisfying (34), and let Mk be the sets defined above. Then:  A) is a homotopy equivalence. (i) The inclusion (M∞ , A) → ( M, (ii) M = M(ρ) := {Mk }k∈Z is a cellular filtration of M∞ , with Wk M = Hk (Mk , Mk−1 )
Ck (X),

∀k ∈ N.

More precisely, the choice of an orientation of each unstable manifold W u (x) determines an isomorphism Θk (ρ): Ck (X)
Wk M(ρ), x → θ∗x (ωk ),

∀x ∈ restk (X),


 where θ x : (Dk , ∂Dk ) → (Mk , Mk−1 ) is a map of the form θ x (ξ) = φ t(ξ), w(ξ) , with w an orientation preserving embedding of Dk onto an open neighbor
 hood of x in W u (x), and 0 ≤ t < t+ so large that φ t(ξ), w(ξ) ∈ Mk−1 for every ξ ∈ ∂Dk . (iii) If ρ ≤ ρ, then the inclusion j = jρ ρ : M∞ (ρ ) → M∞ (ρ) is a cellular homotopy equivalence with respect to the cellular filtrations {Mk (ρ )}k∈Z and {Mk (ρ)}k∈Z . Moreover, the diagram W9 k M(ρ ) s

Θk (ρ ) sss

sss sss

Ck (X)

Θk (ρ)



Wk j

(35)

/ Wk M(ρ)

commutes. By (iii), the isomorphism class of the cellular chain complex Wk M(ρ) does not depend on the choice of the function ρ satisfying (34). In order to fix a standard representative, we can define W∗ (X) := lim W∗ M(ρ), ρ↓0

the limit of the direct system of chain complexes {W∗ M(ρ), W∗ jρ ρ }. The chain complex W∗ (X) is said the Morse complex of X. By Theorem 2.7, the homology of such a chain complex is isomorphic to the singular homology of (M∞ , A), which

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A. ABBONDANDOLO AND P. MAJER

by statement (i) of the theorem above is isomorphic to the singular homology of  A): ( M,  A) ∀k ∈ N. Hk W∗ (X)
Hk ( M, In particular when A is the empty set (so that X is a positively complete Morse vector field on M admitting a Lyapunov function which is bounded below), the homology of the Morse complex is isomorphic to the singular homology of M. By (ii) and by the commutativity of diagram (35), a choice of an orientation of each unstable manifold allows to identify the groups Ck (X) and Wk (X), by the isomorphism Θk = lim Θk (ρ): Ck (X)
Wk (X). ρ↓0

EXERCISE 2.9. Deduce the so called strong Morse relations: there exists a formal series Q with coefficients in N ∪ {+∞} such that ∞  k=0

| restk (X)| tk =

∞ 

 A)tk + (1 + t)Q(t), βk ( M,

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k=0

 A) = rank Hk ( M,  A) ∈ N ∪ {+∞} is the kth Betti number of ( M,  A). where βk ( M, Before proving Theorem 2.8, we recall the semi-continuity properties of the  entrance time function into a subset C ⊂ M: 

tC (p) := inf t ∈ [0, t+ (p)[ φ(t, p) ∈ C ∈ [0, +∞]. LEMMA 2.10. If C is open, tC is upper semi-continuous. If C is closed, tC is lower semi-continuous. Proof. Assume that C is open. It tC (p) < t, there exists s ∈ [tC (p), t[ such that φ(s, p) ∈ C. By continuity, φ(s, q) ∈ C for every q in a neighborhood of p, so tC (q) ≤ s < t in such a neighborhood. Assume that C is closed. If tC (p) > t, choosing t ∈]t, tC (p)[ we have that  \ C for every s ∈ [0, t ]. By continuity and φ(s, p) belongs to the open set M  \ C for every s ∈ [0, t ] and every q in a neighborhood compactness, φ(s, q) ∈ M of p. Therefore, tC (q) ≥ t > t in such a neighborhood.
 either Proof of Theorem 2.8. (i) By Proposition 2.2(i), the orbit of every p ∈ M converges to some rest point x ∈ M for t → +∞, or eventually enters A. Since M∞  the entrance time of is a neighborhood of rest(X) and contains A, for every p ∈ M φ(·, p) in M∞ , 

t M∞ (p) := inf t ∈ [0, t+ (p)[ φ(t, p) ∈ M∞ is finite, and less than t+ (p). Since M∞ is open, by Lemma 2.10 the function t M∞ is upper semi-continuous.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

31

On the other hand, the function t+ is lower semi-continuous. A simple argument with partitions of unity (also known as Dowker theorem; see Dugundji, 1978, VIII.4.3) shows that on a paracompact topological space we can always find a continuous function between an upper semi-continuous function and a lower  → R such that semi-continuous one. So we can find a continuous function s : M + t M∞ < s < t . Then the continuous map  A) → (M∞ , A), r: ( M,

r(p) = φ(s(p), p),

 A), the homotopies is a homotopical inverse of the inclusion i: (M∞ , A) → ( M, id( M,A) ∼ i ◦ r and id(M∞ ,A) ∼ r ◦ i being the map   [0, 1] × A) → ( M,  A), ([0, 1] × M,

(λ, p) → φ(λs(p), p),

and its restriction to ([0, 1] × M∞ , [0, 1] × A) into (M∞ , A). (ii) Let us prove that M is a cellular filtration. Since M is an open covering of M∞ , we just need to compute the singular homology of (Mk , Mk−1 ). Since Mk is the union of the open sets Mk−1 and  
 φ [0, +∞[ × E˚ x ρ(x) , Uk := x∈restk (X)

by excision the singular homology of (Mk , Mk−1 ) is isomorphic to the singular homology of (Uk , Uk ∩ M   k−1 ). Condition (34) implies that the open sets U(x) :=
˚ φ [0, +∞[×E x ρ(x)bigr) , x ∈ restk (X), are pairwise disjoint, so  H∗ (U(x), U(x) ∩ Mk−1 ). H∗ (Mk , Mk−1 )
H∗ (Uk , Uk ∩ Mk−1 )
x∈restk (X)

We shall prove that (U(x), U(x) ∩ Mk−1 ) is homotopically equivalent to a kdimensional disc modulo its boundary, so that  0 if j  k, H j (U(x), U(x) ∩ Mk−1 ) = Z if j = k, proving that M is a cellular filtration. Set for simplicity ρ = ρ(x). By Lemma 1.10(iii), E ux (ρ) × E˚ sx (ρ) ⊂ U(x). Let p ∈ U(x) \ E ux (ρ) × E˚ sx (ρ). By Proposition 2.2(ii), the orbit of p either eventually enters A, or converges to a rest point y for t → +∞. In the latter case, y  x because of Theorem 1.20(i), so by (34) m(y) ≤ k − 1. In both cases, the orbit of p eventually enters Mk−1 . The upper semi-continuous function  0 if p ∈ E˚ x (ρ), a˜ : U(x) → R, p → t Mk−1 (p) if p ∈ U(x) \ E˚ x (ρ),

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A. ABBONDANDOLO AND P. MAJER

is strictly less than the lower semi-continuous function t+ , so we can find a continuous function a: U(x) → [0, +∞[ such that a˜ < a < t+ , so that φ(a(p), p) ∈ Mk−1

∀p ∈ U(x) \ E˚ x (ρ).

Then we can define the continuous map 
α: E ux (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) → (U(x), U(x) ∩ Mk−1 ),

(37)

p → φ(a(p), p).

By Theorem 1.20(i), for every p ∈ U(x) there holds



 b(p) := sup t ∈ ]t− (p), 0] φ(t, p) ∈ E˚ x (ρ) = max t ∈ ]t− (p), 0] φ(t, p) ∈ E x (ρ) , so by Lemma 2.10 the function b: U(x) → ]−∞, 0] is both lower and upper semicontinuous, hence continuous. The map p → φ(b(p), p) is the identity on E ux (ρ) × E˚ sx (ρ), and maps all the other points of U(x) into ∂E ux (ρ) × E˚ sx (ρ). Since by (34) E x (ρ) ∩ Mk−1 = ∅, the continuous map
 β: (U(x), U(x) ∩ Mk−1 ) → E ux (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) , p → φ(b(p), p), is well-defined. It is easy to check that α and β are homotopy inverses. Indeed,
 (λ, p) → β φ(λa(p), p) is a homotopy between β ◦ α and the identity map on 
u E x (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) . On the other hand, by (37),

  (λ, p) → φ a φ(λb(p), p) , φ(λb(p), p) is a homotopy between α ◦ β and the map (U(x), U(x) ∩ Mk−1 ) → (U(x), U(x) ∩ Mk−1 ),

p → φ(a(p), p),

which is clearly homotopy equivalent to the identity on (U(x), U(x) ∩ Mk−1 ). We conclude that (U(x), U(x) ∩ Mk−1 ) is homotopy equivalent to
u  E x (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) , 
which is homotopy equivalent to E ux (ρ), ∂E ux (ρ) , a k-dimensional disc modulo
u u (x), and the its boundary. The latter pair is homeomorphic to Wloc,ρ (x), ∂Wloc,ρ statement about the form of the isomorphism Θk easily follows. (iii) Since Mk (ρ ) ⊂ Mk (ρ) for every k, j is a cellular map; we will construct a cellular homotopy inverse of j of the form γ(p) = φ(c(p), p),

(38)

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

33

with c a suitable positive continuous function. Given p ∈ M∞ (ρ), set 

κ(p) := min k ∈ N p ∈ Mk (ρ) , and c˜ (p) := t Mκ(p) (ρ ) (p), the entrance time of p into the open set Mκ(p) (ρ ). By (34), every point in Mk (ρ) either eventually enters A or converges to a rest point x with m(x) ≤ k; in both cases, p eventually enters Mk (ρ ). Therefore, c˜ < t+ . Since {Mh (ρ )}h∈Z is a filtration, t Mh (ρ ) is nonincreasing in h, so 



c˜ (p) = min t Mh (ρ ) (p) 0 ≤ h ≤ κ(p) = min t Mh (ρ ) (p)χh (p) h ∈ N , where χh (p) = 1 if h ≤ κ(p), i.e., p  Mh−1 (ρ), and χh (p) = +∞ otherwise; hence the positive function χh is upper semi-continuous. Since also t Mh (ρ ) is upper semi-continuous and nonnegative, so is the function c˜ . Let c: M∞ (ρ) → R be a continuous function such that c˜ < c < t+ , and let γ: M∞ (ρ) → M∞ (ρ ) be the map defined in (38). By construction, γ maps Mk (ρ) into Mk (ρ ), so it is a cellular map. The cellular homotopies id M∞ (ρ) ∼ j ◦ γ and id M∞ (ρ ) ∼ γ◦ j are given by the cellular map (λ, p) → φ(λc(p), p) on the respective domains. If
 θ x (ρ): (Dk , ∂Dk ) → Mk (ρ), Mk−1 (ρ) and


 θ x (ρ ): (Dk , ∂Dk ) → Mk (ρ ), Mk−1 (ρ )

are the continuous maps appearing in (ii), then j ◦ θ x (ρ ) is homotopic to θ x (ρ), so the diagram (35) commutes.


2.8. REPRESENTATION OF ∂∗ IN TERMS OF INTERSECTION NUMBERS

Let us strengthen the Morse – Smale assumption (A7) by requiring: (A7 ) X satisfies the Morse – Smale condition up to order 1. In this case, the boundary operator ∂k of the Morse complex of X can be expressed in terms of intersection numbers of unstable and stable manifolds of rest points of index difference 1. First of all notice that if m(x) − m(y) = 1, the assumption (A7 ) implies that W u (x) ∩ W s (y) is a flow-invariant 1-dimensional manifold, that is a discrete set of flow lines. We claim that W u (x)∩W s (y) is compact: otherwise Corollary 2.4 would imply the existence of a “broken orbit” from z0 = x to zh = y, with intermediate

34

A. ABBONDANDOLO AND P. MAJER

rest points z1 , . . . , zh−1 , for some h ≥ 2. By the Morse – Smale condition (up to order 0) m(z0 ) > m(z1 ) > · · · > m(zh ), a contradiction because m(z0 ) − m(zh ) = 1. Therefore W u (x) ∩ W s (y) consists of finitely many flow lines. Let us fix an orientation of each unstable manifold W u (x). As we have seen in Section 2.7, this choice determines a preferred isomorphism Θk : Ck (X)
Wk (X). Moreover, it determines an orientation of each transverse intersection W u (x) ∩ W s (y). Indeed, the orientation of each unstable manifold determines a coorientation of each stable manifold (that is an orientation of its normal bundle), and the transverse intersection of a finite-dimensional oriented submanifold with a finite-codimensional co-oriented submanifold carries a canonical orientation: if p ∈ W u (x)∩W s (y) and V ⊂ T p W u (x) is a linear complement of T p (W u (x)∩W s (y)) in T p W u (x), by transversality V is also a complement of T p W s (x) in T p M, so it is oriented, and the orientation of W u (x) ∩ W s (y) is the one for which 
T p W u (x) = T p W u (x) ∩ W s (y) ⊕ V is an oriented sum. In particular, if m(x)−m(y) = 1 each connected component W of W u (x)∩W s (y) is an oriented line, and we can define (W) to be +1 if φ is orientation preserving on W, −1 otherwise. Then we can define the integer  n(x, y) = (W), ∀x, y ∈ rest(X), m(x) − m(y) = 1. W connected component of W u (x) ∩ W s (y)

Assume that conditions (A1) – (A6), (A7 ), and (A8) hold. Then we have the following fact. THEOREM 2.11. In terms of the preferred isomorphism Θk : Ck (X)
Wk (X), the boundary operator of the Morse complex of X has the form  ∂k x = n(x, y)y, ∀x ∈ restk (X) ⊂ Ck (X). (39) y∈restk−1 (X)

Before proving this result, we recall that if σn denotes the generator of Hn (S n ) corresponding to the standard orientation of ∂Dn+1 = S n , that is the one for which Rn+1 = Rζ ⊕T ζ S n is an oriented sum, for every ζ in S n , we have that the boundary homomorphism Hn+1 (Dn+1 , ∂Dn+1 ) → Hn (∂Dn+1 ) maps ωn+1 into σn . EXERCISE 2.12. Let A1 , . . . , Ah be pairwise disjoint closed n-discs in S n , with maps h   ai : (Dn , ∂Dn ) → S n , S n A˚ i , i=1

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

35

mapping Dn homeomorphically onto Ai , preserving the standard orientations. Let  j: S n → (S n , S n \ hi=1 A˚ i ) be the inclusion. Then j∗ (σn ) =

h 

ai∗ (ωn ).

i=1

Proof of Theorem 2.11. Notice first of all that by (A2) and (A6), for every x ∈ rest(X) there are finitely many rest points y ∈ rest(X) with f (y) < f (x), so the sum appearing in (39) is finite. Let ρ: rest(X) → ]0, +∞[ be a function satisfying (34), and let Mk = Mk (ρ), for k ∈ N ∪ {∞}. Let us fix a rest point x of Morse index k. By the naturality of the boundary homomorphism of pairs in singular homology, we have the commutative diagram Hk (Dk , ∂Dk ) 

Hk−1 (∂Dk )

θ∗x

α∗

/ Hk (Mk , Mk−1 )  / Hk−1 (Mk−1 )

where α: ∂Dk → Mk−1 is the restriction of θ x . The cellular boundary homomorphism ∂k of the cellular filtration {Mk }k∈Z is the composition of the right vertical arrow with the homomorphism induced by the inclusion i: Mk−1 → (Mk−1 , Mk−2 ). On the other hand, the left vertical arrow is an isomorphism mapping ωk into σk−1 . Therefore, ∂k maps the generator θ∗x (ωk ) of Hk (Mk , Mk−1 ) into i∗ α∗ (σk−1 ) ∈ Hk−1 (Mk−1 , Mk−2 ), and we must express the latter element in terms y of the generators θ∗ (ωk−1 ) of Hk−1 (Mk−1 , Mk−2 ), for y ∈ restk−1 (X). By the Morse – Smale condition up to order 1,   −1 s α W (y) = {ζ1 , . . . , ζh } y∈restk−1 (X)

is a finite subset of ∂Dk , and α maps all the other points into points which either belong to stable manifolds of rest points of index less than k − 1, or eventually enter A; so the orbit of any point in α(∂Dk \ {ζ1 , . . . , ζh }) eventually enters Mk−2 . Choose r > 0 so small that the closed r-balls Br (ζi ) ⊂ ∂Dk centered in ζi are pairwise disjoint (k − 1)-discs. Let b: ∂Dk → R be a continuous function such that χt Mk−2 ◦ α < b < t+ ◦ α,  where χ is the characteristic function of the open set ∂Dk \ hi=1 Br (ζi ), and t Mk−2 is the entrance time function into Mk−2 . Then α is homotopic to the map
 β: ∂Dk → Mk−1 , ζ → φ b(ζ), α(ζ) ,

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A. ABBONDANDOLO AND P. MAJER

so θ∗x (ωk ) = i∗ α∗ (σk−1 ) = i∗ β∗ (σk−1 ). Denote by γi : (Dk−1 , ∂Dk−1 ) → (Mk−1 , Mk−2 ) the composition of i ◦ β with an orientation preserving homeomorphism
 (Dk−1 , ∂Dk−1 ) → Bρ (ζi ), ∂Bρ (ζi ) . Then the result of Exercise 2.12 shows that: θ∗x (ωk )

= i∗ β∗ (σk−1 ) =

h 

γi ∗ (ωk−1 ).

(40)

i=1

Fix some i ∈ {1, . . . , h}, let y be the rest point of index k − 1 toward which the orbit of α(ζi ), i.e., of β(ζi ), converges for t → +∞, and let Wi be the connected component of W u (x) ∩ W s (y) consisting of such an orbit. We claim that γi is homotopic to either θy , in the case (Wi ) = 1, or to θy ◦ µ, where µ is an orientation reversing automorphism of (Dk−1 , ∂Dk−1 ), in the case (Wi ) = −1. Therefore y

γi ∗ (ωk−1 ) = (Wi )θ∗ (ωk−1 ), and (40) allows to conclude. Let us proof the claim. Up to a small perturbation, we may assume that γi is a C 1 embedding of a closed (k − 1)-disc, meeting W s (y) transversally at the single point p = γi (0). The diffeomorphism γi induces an orientation of T p γi (Dk−1 ), the one for which T p W u (x) = RX(p) ⊕ T p γi (Dk−1 ) is an oriented sum. The differential of the flow D2 φ(t, ·) at p maps the tangent space of γi (Dk−1 ) at p onto a subspace of T φ(t,p) M which converges to T y W u (y) for t → +∞ (see for instance Abbondandolo and Majer, 2003c, Theorem 2.1(iii)). A first consequence is that the orientation of T p γi (Dk−1 ) defined above is (Wi ) times the orientation obtained by seeing T p γi (Dk−1 ) as a complement of T p W s (y) in T p M. A second consequence is that, by the evolution of graphs of Lipschitz maps from Eyu (r) to Eys (r) near the hyperbolic rest point y (see Shub, 1987, or Abbondandolo and Majer, 2001, Proposition A.3 and Addendum A.5), if r > 0
 is small and t ≥ 0 is large then φ {t} × γi (Dk−1 ) ∩ Ey (r) is the graph of a map3 τ : Eyu (r) → Eys (r). Let K ⊂ Dk−1 be the closed neighborhood of 0 such that φ({t} × γi (K)) = graph τ. 3 This statement is part of the content of the so called λ-lemma, in the particular case of a gradient-like flow. See Palis (1968) and Palis and de Melo (1982).

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

37

Since K is a closed (k − 1)-disc, it is a deformation retract of Dk−1 . Since the local u (y) is also the graph of a map σu : E u (r) → E s (r), it is unstable manifold Wloc,r y y now easy to combine the above maps to construct a homotopy between γi and an embedding of (Dk−1 , ∂Dk−1 ) into (W u (y), W u (y) ∩ Mk−2 ), which is orientation preserving, hence homotopic to θy , if (Wi ) = 1, orientation reversing, hence homotopic to θy ◦ µ, if (Wi ) = −1.


2.9. HOW TO REMOVE THE ASSUMPTION (A8)

If we drop assumption (A8), there need not exist a function ρ satisfying (34), and it becomes more difficult to associate a cellular filtration to X. Nevertheless, we can make the graded group C∗ (X) into a chain complex by taking a direct limit of the Morse complexes on sublevels { f < a}, for a ↑ sup f . On these domains indeed, there are finitely many rest points and condition (A7) guarantees condition (A8). Not being forced to assume (A8) is a positive fact, in that assumption (A7) can be more easily achieved by generic perturbations, as we shall see in Section 2.12. If the supremum of f on M is attained, by (A2) and (A6) X has finitely many rest points, so (A8) is implied by (A7). Thus, we can assume that sup f is not attained. a := A ∪ { f < For a < sup f , let W∗ (X)a be the Morse complex associated to M a}, and if a < b < sup f , let wab : W∗ (X)a → W∗ (X)b a → M b . The Morse complex of X be the chain map induced by the inclusion M is defined to be the chain complex W∗ (X) := lim W∗ (X)a , a↑sup f

the limit of the direct system {W∗ (X), wab }. Notice that if (A8) holds, so that W∗ (X) is the chain complex defined in Section 2.7, the family of chain complexes {W∗ (X)a }a<sup f is identified with an increasing and exhausting family of sub-complexes of W∗ (X), so this definition of the Morse complex agrees with the previous one. Since the homology of a direct limit of chain complexes is the direct limit of the homologies (see Dold, 1980, VIII.5.20), Hk W∗ (X) = lim Hk W∗ (X)a . a↑sup f

Similarly, the singular homology of an increasing union of open subsets is the limit of the singular homologies (see Dold, 1980, VIII.5.22), so  A) = lim Hk ( M a , A). Hk ( M, a↑sup f

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A. ABBONDANDOLO AND P. MAJER

We conclude that the homology of the Morse complex of X is isomorphic to the  A), singular homology of ( M, 
 A) ∀k ∈ N. Hk W∗ (X)
Hk ( M, Finally, having fixed an orientation for each unstable manifold, we have the isomorphisms Θak : Ck (X)a
Wk (X)a , Ck (X)a being the subgroup of Ck (X) generated by the rest points x with f (x) < a, and the limit of this direct system defines an isomorphism Θk : Ck (X)
Wk (X). REMARK 2.13. Since the boundary of x ∈ rest(X) in C∗ (X) and in C∗ (X)a coincide when f (x) < a < sup f , the formula for the boundary homomorphism under the Morse – Smale condition up to order 1 (Theorem 2.11) holds also without assuming (A8).

2.10. MORSE FUNCTIONS ON HILBERT MANIFOLDS

A particular but important case is the following situation: f is a C 2 Morse function on a smooth Hilbert manifold N, endowed with a C 1 Riemannian metric g, −∞ <  = {p ∈ N | f (p) < b}, M = {p ∈ N | a < f (p) < b}, and X = a < b ≤ +∞, M −∇ f , the negative gradient of f with respect to the metric g. Let us see what the assumptions (A1) – (A8) look like in this situation. In this case, of course, rest(X) = crit( f ) ∩ {a < f < b}, the set of critical points of f with values between a and b. Condition (A2) is equivalent to saying that f is a Morse function on M, and in condition (A3) the Morse index is the standard Morse index of a critical point of f | M . The set of critical points of f with index k will be denoted by critk ( f ). Condition (A4) is automatically fulfilled, f itself being a Lyapunov function for −∇ f , and so is condition (A5). In the case of a gradient flow the (PS) condition can be restated in the more familiar way: the pair ( f, g) satisfies the (PS) condition at level c ∈ R if every  such that f (pn ) → c and D f (pn ) → 0 is compact (here the sequence (pn ) ⊂ M ∗ norm · on T M is induced by the Riemannian structure g). The assumption (A6) is equivalent to: ( f, g) satisfies the (PS) condition at level c for every c ∈ [a, b[, and a is a regular value for f . As we shall see in Section 2.12, the Morse – Smale condition required in (A7) can be always achieved by perturbing the metric g.  g) is complete. Finally, assumption (A1) is automatically fulfilled when ( M, Indeed, the following fact holds.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

39

 R) and a ∈ R be such that the strip PROPOSITION 2.14. Let f ∈ C 2 ( M,  induced {a ≤ f ≤ c} is complete (with respect to the geodesic distance d on M by the Riemannian metric g), for every c < sup f . Then the vector field −∇ f is positively complete with respect to { f < a}.  and consider the curve u: [0, t+ (p)[ → M,  u(t) = φ(t, p). If Proof. Let p ∈ M + f (p) = sup f , then p is a critical point of f , so t (p) = +∞. If inf f ◦ u < a then u(t) eventually enters { f < a}. Therefore we can assume that f (p) < sup f and u([0, t+ (p)[) ⊂ {a ≤ f ≤ f (p)}, and we must prove that t+ (p) = +∞. Let 0 ≤ s < t. Then  t   t





 D f u(τ) −∇ f u(τ) dτ = − g ∇ f u(τ) , ∇ f u(τ) dτ, f u(t) − f u(s) = s

s

and the Cauchy – Schwarz inequality implies that  t

 g u (τ), u (τ) dτ d u(s), u(t) ≤ s t  

 = g ∇ f u(τ) , ∇ f u(τ) dτ s  t 
√ 
 1/2 ≤ t−s g ∇ f u(τ) , ∇ f u(τ) dτ s  √

 √ = t − s f u(s) − f u(t) ≤ t − s f (p) − inf f ◦ u. The above estimate shows that u is uniformly continuous. If by contradiction t+ (p) < +∞, by the completeness of the strip {a ≤ f ≤ f (p)} we deduce that u(t) converges for t → t+ (p). But then the solution u of u = −∇ f (u), u(0) = p, can be extended to a right neighborhood of t+ (p), contradicting the maximality of t+ (p).
We summarize the above discussion into the following proposition. PROPOSITION 2.15. Let f be a C 2 function on the smooth Hilbert manifold N, endowed with a C 1 Riemannian metric g, and let −∞ < a < b ≤ +∞. Assume that (B1) a is a regular value of f ; (B2) f is a Morse function on {a < f < b}, and it has only critical points of finite Morse index in such a strip; (B3) for every c < b, the strip {a ≤ f ≤ c} is complete; (B4) f satisfies the (PS) condition at every level c ∈ [a, b[.  = { f < b}, X  = −∇ f |  and M = {a < f < b}, the conditions Then, setting M M (A1) – (A6) are fulfilled.

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A. ABBONDANDOLO AND P. MAJER

Notice that only (B3) and (B4) involve the metric. Moreover, if (B3) and (B4) hold for some metric, they hold also for every uniformly equivalent metric. Under the assumptions (B1) – (B4), the free Abelian group generated by the critical points of f of index k in {a < f < b} will be denoted by Ck ( f )ba . The lower index will be omitted when a < inf f , the upper index will be omitted when b = +∞. If −∇ f satisfies also the Morse – Smale condition up to order 0 on {a < f < b}, the boundary operator of the Morse complex of −∇ f on {a < f < b} will be denoted by ∂k ( f, g)ba : Ck ( f )ba → Ck−1 ( f )ba . Its homology is isomorphic to the singular homology of ({ f < b}, { f < a}): Hk ({C∗ ( f )ba , ∂∗ ( f, g)ba })
Hk ({ f < b}, { f < a}).

2.11. BASIC RESULTS IN TRANSVERSALITY THEORY

In the following lemma we single out a useful family of linear mappings whose kernel is complemented. LEMMA 2.16. Let E, F, G be Banach spaces, and assume that A ∈ L(E, G) has complemented kernel and finite-codimensional range. Then for every B ∈ L(F, G) the kernel of the operator C ∈ L(E × F, G), C(e, f ) = Ae − B f , is complemented in E × F. Proof. Let E0 := ker A, E1 be a closed complement of E0 in E, and P0 , P1 be the associated projectors. Let G1 := ran A, G0 be a (finite-dimensional) complement of G1 in G, and Q0 , Q1 be the associated projectors. Then A induces an isomorphism from E1 onto G1 , whose inverse will be denoted by T ∈ L(G1 , E1 ). The equation C(e, f ) = 0 is equivalent to AP1 e = B f , which is equivalent to the system  AP1 e = Q1 B f, Q0 B f = 0, again equivalent to



P1 e = T Q1 B f, Q0 B f = 0.

(41)

Since Q0 B has finite rank, its kernel — say F0 — has a (finite-dimensional) complement F1 . By (41), the kernel of C is 

ker C = (e0 + T Q1 B f0 , f0 ) ∈ E × F e0 ∈ E0 , f0 ∈ F0 , and the closed linear subspace E1 × F1 is a complement of ker C.




MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

41

Let us recall some definitions and basic facts about transversality in a Banach setting. A classical reference for these topics is Abraham and Robbin (1967). If ϕ: M → N is a C k map between Banach manifolds, k ≥ 1, a point q ∈ N is said a regular value for ϕ if for every p ∈ ϕ−1 ({q}) the differential Dϕ(p): T p M → T q N is a left inverse, i.e., if it is onto and its kernel is complemented. In this case, ϕ−1 ({q}) is a submanifold of class C k . A C 1 map ϕ: M → N between Banach manifolds is said a Fredholm map if its differential at every point is a Fredholm operator. When the index of the differential is constant (for instance when M is connected), this integer is said the Fredholm index of ϕ. PROPOSITION 2.17. Let M, N, O be Banach manifolds, and let ϕ ∈ C 1 (M, N), ψ ∈ C 1 (M, O) be maps with regular values p ∈ N and q ∈ O. Then: (i) p is a regular value for ϕ|ψ−1 ({q}) if and only if q is a regular value for ψ|ϕ−1 ({p}) ; (ii) ϕ|ψ−1 ({q}) is a Fredholm map if and only if ψ|ϕ−1 ({p}) is a Fredholm map, in which case the indices coincide. This proposition is a consequence of the following linear statements. PROPOSITION 2.18. Let E, F, G be Banach spaces, and let A ∈ L(E, F), B ∈ L(E, G) be left inverses. Then: (i) A|ker B is a left inverse if and only if B|ker A is a left inverse; (ii) A|ker B is Fredholm if and only if B|ker A is Fredholm, in which case the indices coincide. Proof. Let R ∈ L(F, E) and S ∈ L(G, E) be right inverses of A and B, respectively. (i) If R0 ∈ L(F, ker B) is a right inverse of A|ker B , i.e., a right inverse of A with range in ker B, the map S 0 := (IE − R0 A)S is a right inverse of B, being a perturbation of S by an operator with range in ker B, and it takes value in ker A because AS 0 = AS − AR0 AS = AS − IF AS = 0. Therefore, S 0 is a right inverse of B|ker A . (ii) The kernels of A|ker B and B|ker A coincide: ker A|ker B = ker B|ker A = ker A ∩ ker B. Moreover, since R: F → RF is an isomorphism and since I − RA is a projector onto ker A, coker A|ker B =

R ker A + RF E F RF
= .
A ker B RA ker B ker A + RA ker B ker A + ker B

42

A. ABBONDANDOLO AND P. MAJER

We conclude that the assertions in (ii) are equivalent, each of them being equivalent to the fact that the pair of subspaces (ker A, ker B) is Fredholm, i.e., ker A ∩ ker B is finite-dimensional, and ker A + ker B is finite-codimensional.4 The index of A|ker B and of B|ker A equals the index of (ker A, ker B), ind(ker A, ker B) = dim ker A ∩ ker B − codim(ker A + ker B).




We recall that a subspace T of a topological space T is said residual if it contains a countable intersection of open and dense subspaces of T . Baire theorem guarantees that a residual subspace of a complete metric space is dense. The following Sard – Smale Theorem, combined with Proposition 2.17, is the basic tool to deal with transversality questions. THEOREM 2.19. Let M, N be C h Banach manifolds, h ≥ 1, with M Lindel¨of. Let ϕ: M → N be a C h Fredholm map of index m. If h > max{0, m} then the set of regular values of ϕ is residual in N. The proof can be found in Smale (1965). 2.12. GENERICITY OF THE MORSE – SMALE CONDITION

Let f be a C h+1 real function, h ≥ 1, on the smooth Hilbert manifold N, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B1) – (B4). The aim of this section is to show that it is possible to perturb the metric g obtaining a uniformly equivalent metric such that the associated negative gradient of f has the Morse – Smale property up to order h. We shall assume N to be infinite-dimensional and second countable (in particular, it is modeled on a separable Hilbert space). A well-known theorem by Eells and Elworthy (1970) implies that every infinite-dimensional Hilbert manifold can be smoothly embedded as an open subset of a Hilbert space. So we may assume that N is an open subset of the separable Hilbert space (H, ·, ·).5 Denote by Sym(H) the Banach space of self-adjoint bounded linear operators on H. The metric g can be represented by a C h map G: N → Sym(H) taking values in the cone of positive operators, such that g(p)[ξ, η] = G(p)ξ, η ∀p ∈ N, ∀ξ, η ∈ T p N = H. 4

See also Section 3.2. Viewing N as an open subset of a Hilbert space is useful to simplify the notation (some spaces of maps are Banach spaces and not Banach manifolds, some sections of Banach bundles are just maps between Banach spaces, and so on) but it is by no means necessary. Therefore the results of this section hold also for a finite-dimensional N which is not diffeomorphic to an open subset of Rn . 5

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

43

We shall always denote by a lower case letter a symmetric bilinear form, and by the corresponding upper case letter the associated self-adjoint operator. The gradient of f with respect to the metric g is ∇g f (p) = G(p)−1 ∇ f (p), where ∇ f denotes the gradient of f with respect to the Hilbert inner product ·, ·. The Morse – Smale property will be achieved by rank 2 perturbations of G. In order to describe the space of such perturbations, let θ: N → [0, +∞[ be a continuous function such that θ(p) ≤

1 G(p)−1 

∀p ∈ N.

(42)

The vector space
  K := K ∈ Cbh N, Sym(H) rank K(p) ≤ 2 ∀p ∈ N,

∃c ≥ 0 such that K(p) ≤ c θ(p) ∀p ∈ N is a Banach space with the norm K(p) . θ(p)0 θ(p)

KK := KC h + sup

As usual, the symbol Cbh denotes the space of maps whose differentials up to the hth order are continuous and bounded. Notice that the maps K ∈ K vanish on the set of zeroes of θ. By (42), for every p ∈ N G(p)−1 K(p) ≤ G(p)−1 KK θ(p) ≤ KK , so if KK < 1, G + K = G(I + G−1 K) is positive, and defines a metric g + k which is uniformly equivalent to g. Denote by K1 the open unit ball of K. The main result of this section is the following theorem. THEOREM 2.20. Let f be a C h+1 function, h ≥ 1, on the smooth second countable Hilbert manifold N ⊂ H, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B2). Assume that the continuous function θ: N → [0, +∞[ satisfies (42), that its set of zeroes is the closure of an open set, and that it has the following property: if x, y are critical points in {a < f < b} with m(x) − m(y) ≤ h, such that W u (x) and W s (y) (with respect to −∇g f ) have a nontransverse intersection at p, then θ > 0 somewhere on the orbit of p. Then for every K in a residual subspace of K1 , the metric g + k associated to G + K is such that the vector field −∇g+k f satisfies the Morse – Smale property up to order h. Notice that high regularity of f and g is needed if we want to achieve the Morse – Smale property up to a high order. This phenomenon is determined by

44

A. ABBONDANDOLO AND P. MAJER

the regularity versus Fredholm index assumption required by the Sard – Smale Theorem 2.19. In a finite-dimensional setting this problem does not occur because there C h functions can always be C h -approximated by smooth ones, while such an approximation may not be possible on an infinite-dimensional Hilbert space (see for instance Nemirovski˘ı and Semenov, 1973 and Lasry and Lions, 1986). Notice that C 2 regularity of f is enough to get the Morse – Smale property up to order 1, which is just what we need in order to have the Morse complex and to represent it by intersection numbers. The possibility of having a function θ which vanishes on some regions where the intersections are already transversal and which can be very small elsewhere will be useful in Section 2.13. Let us set up the proof of Theorem 2.20. Fix two critical points x  y in {a < f < b} with m(x) − m(y) ≤ h, and consider the space of curves

 C = C(x, y) := u ∈ C 1 (R, N) lim u(t) = x, lim u(t) = y, lim u (t) = 0 . t→−∞ t→+∞ t→±∞ The space C is a smooth Banach manifold, being an open subset of an affine Banach space modeled on C01 (R, H) (the spaces C0h are defined in Section 1.2). Therefore, T u C = C01 (R, H). The map Ψ: C × K1 → C00 (R, H),

(u, K) → u + ∇g+k f (u) = u + (G + K)−1 (u)∇ f (u),

is of class C h , and its zeroes are the pairs (u, K) such that u is a negative gradient flow line of f with respect to the metric g+k, going from x to y. Set Z := Ψ−1 ({0}). The following two lemmas describe some properties of the differential of Ψ with respect to the first, respectively the second variable. LEMMA 2.21. Let (u, K) ∈ Z. Then: (i) the operator D1 Ψ(u, K): T u C → C00 (R, H) is Fredholm of index m(x) − m(y); (ii) the operator D1 Ψ(u, K) is onto if and only if the unstable manifold of x and the stable manifold of y with respect to the vector field −∇g+k f intersect transversally at u(t) for some (hence all) t ∈ R; (iii) if w ∈ C00 (R, H) and a < b are real numbers, then there exists v ∈ T u C such that D1 Ψ(u, K)[v](t) = w(t) ∀t ∈ ]−∞, a] ∪ [b, +∞[. Proof. The differential of Ψ with respect to the first variable is of the form D1 Ψ(u, K): C01 (R, H) → C00 (R, H),

v → v − Av,

where A: R → L(H) is defined by



 A(t) := −(G + K)−1 u(t) D2 f u(t) − D(G + K)−1 ) u(t) ∇ f u(t) .

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

45

Since u(t) converges to x, resp. to y, for t → −∞, resp. t → +∞, A(t) converges in norm to the operators A(−∞) = −(G + K)−1 (x)D2 f (x) = −∇2g+k f (x), A(+∞) = −(G + K)−1 (y)D2 f (y) = −∇2g+k f (y), which are hyperbolic, and have positive eigenspaces of dimension m(x) and m(y), respectively. Then (i) follows from Proposition 1.8. Claim (ii) follows from the second identity in (9) , and from the identities T u(0) W u (x) = WAu ,

T u(0) W s (y) = WAs .

As for claim (iii), up to a translation we may assume that a < 0 < b. Then the conclusion follows from Proposition 1.6(i), applied to A|[0,+∞] and to A|[−∞,0] (−·).

 LEMMA 2.22. Let (u, K) ∈ Z, and let a < b be real numbers such that θ u(t)  0 for every t ∈ [a, b]. Let w ∈ C h (R, H) be a curve with support in [a, b]. Then there exists J ∈ K such that D2 Ψ(u, K)[J] = w. Proof. The differential of Ψ with respect to the second variable is D2 Ψ(u, K)[J] = −(G + K)−1 (u)J(u)(G + K)−1 (u)∇ f (u). Since (u, K) ∈ Z, the curve u is a flow line of the vector field −∇g+k f going from x to y. In particular, u is a C h+1 embedding of R into N, and ∇ fg+k ◦u never vanishes. It is easy to find a C h curve J0 : R → Sym(H) with support in [a, b] such that for every t ∈ R the symmetric operator J(t) has rank not exceeding 2, and


 maps the nonzero vector (G + K)−1 u(t) ∇ f u(t) = ∇g+k f u(t) into the vector
 −(G + K) u(t) w(t). Indeed, one may write an explicit formula for J0 by noticing that if ξ  0 and η are two elements of H, the bounded linear operator on H ζ →

ξ, ζ η, ζ ξ, η ξ, ζ η+ ξ− ξ, 2 2 |ξ| |ξ| |ξ|4

is self-adjoint, has rank not exceeding 2, vanishes when η = 0, maps ξ into η, and depends smoothly on (ξ, η) ∈ (H \ {0}) × H. Since u is a C h+1 embedding, given δ > 0 we can find an open neighborhood U
 of u(]a − δ, b + δ[) and a C h+1 submersion τ: U → ]a − δ, b + δ[ such that τ u(t) = t for every t ∈ ]a − δ, b + δ[. Since θ is positive on u([a, b]), up to choosing a smaller δ and a smaller U we may assume that inf U θ > 0, and also that τ has bounded derivatives up to order h+1. If ψ ∈ Cb∞ (H, R) is a cut-off function with support in U
 and taking value 1 on u([a, b]), the C h map J: N → Sym(H), J(p) = ψ(p)J0 τ(p) , belongs to K and has the required property.
The following lemma is the key point in the proof of Theorem 2.20.

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A. ABBONDANDOLO AND P. MAJER

LEMMA 2.23. Let (u, K) ∈ Z. Then the differential DΨ(u, K): T u C × K → C00 (R, H) is a left inverse. Proof. We must prove that the operator DΨ(u, K)[(v, J)] = D1 Ψ(u, K)[v] + D2 Ψ(u, K)[J] is onto and that its kernel is complemented in T u C × K. By Lemma 2.21(i), the operator D1 Ψ(u, K) is Fredholm, so Lemma 2.16 implies that ker DΨ(u, K) is complemented in T u C × K. Moreover, the range of DΨ(u, K) contains the range of D1 Ψ(u, K), in particular it has finite codimension. If θ ◦ u(t) = 0 for every t ∈ R, also K ◦ u vanishes identically, so ∇g+k f ◦ u = ∇g f ◦ u, and (recalling that the set of zeroes of θ is the closure of an open set) the tangent spaces of the unstable and stable manifolds of x and y along u are the same for −∇g+k f and for −∇g f . Therefore, the assumption of Theorem 2.20 guarantees that these manifolds meet transversally along u. By Lemma 2.21(ii), D1 Ψ(u, K) is onto, and so is DΨ(u, K). If θ ◦ u is not identically zero, we can find real numbers a < b such that
 θ u(t)  0 for every t ∈ [a, b]. Let w ∈ C00 (R, H) and let  > 0. By Lemma 2.21(iii), there exists v ∈ T u C such that D1 Ψ(u, K)[v](t) = w(t)

∀t ∈ ]−∞, a] ∪ [b, +∞[.

The curve w − D1 Ψ(u, K)[v] is continuous and has support in [a, b], and we can find a C h curve z : R → H with support in [a, b] such that z − (w − D1 Ψ(u, K)[v])∞ < . Since z has support in [a, b], where θ ◦ u does not vanish, by Lemma 2.22 there exists J ∈ K such that D2 Ψ(u, K)[J] = z. Hence DΨ(u, K)[(v, J)] − w∞ = D1 Ψ(u, K)[v] + z − w∞ < . Therefore, DΨ(u, K) has dense and finite-codimensional range, so it is onto.




In particular, Z is a C h submanifold of C × K1 . Let π be the restriction to Z of the projection onto the second factor in the product C × K1 . LEMMA 2.24. The map π: Z → K1 is Fredholm of index m(x) − m(y). Proof. Everything follows from Proposition 2.17(ii), applied to M = C × K1 , N = K1 , O = C00 (R, H), ϕ: C×K1 → K1 projection onto the second factor, ψ = Ψ, together with Lemma 2.21(i) and Proposition 2.23.
Denote by H(x, y) the set of regular values of π. LEMMA 2.25. The set H(x, y) is residual in K1 .

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

47


 Proof. Notice that R acts freely on the submanifold Z by t, (u, K) → (u(t + ·), K), and the map π is invariant with respect to this action. Therefore, the  → K1  = Z/R is still a C h manifold, and the induced map  π: Z quotient space Z h is of class C and Fredholm index m(x) − m(y) − 1 ≤ h − 1, by Lemma 2.24. Moreover, K is a regular value for π if and only if it is a regular value for  π. Since N, and thus H, is assumed to be second countable, C × K1 is second  Since the level of differentiability of  countable, and so are Z and Z. π is strictly greater than its Fredholm index, the Sard – Smale Theorem 2.19 implies that the set of regular values of  π — and thus of π — is residual in K1 .
Proof of Theorem 2.20. By Proposition 2.17(i), H(x, y) is also the set of K ∈ K1 for which the map Ψ(·, K) : C(x, y) → C00 (R, H) has 0 as a regular value. By Lemma 2.21 (ii), H(x, y) is also the set of K ∈ K1 such that the unstable manifold of x and the stable manifold of y with respect to −∇g+k f meet transversally. By Lemma 2.25, the countable intersection  H(x, y) x, y ∈ crit( f ) ∩ {a < f < b} x  y, m(x) − m(y) ≤ h

is the required residual subset of K1 .




2.13. INVARIANCE OF THE MORSE COMPLEX

Let f ∈ C 2 (M) be a Morse function on the smooth second countable Hilbert manifold M, with critical points of finite index. Assume that f is bounded below and that M admits a complete Riemannian metric g such that ( f, g) satisfies the Palais – Smale condition. We know from the previous section that by perturbing g we can achieve also the Morse – Smale property up to order 1. In general, different Morse – Smale metrics will produce different Morse complexes: the groups Ck ( f ) are the same, but the boundary operators ∂k may vary. Of course the homology of the Morse complex does not vary, being isomorphic to the singular homology of M, but we can say more: varying the metric we obtain isomorphic chain complexes. This fact was observed by Cornea and Ranicki (2003) (together with other interesting rigidity results) for finite-dimensional manifolds, and for some cases of Floer theory. The proof we give here in our infinite-dimensional situation uses an idea from Abbondandolo and Majer (2001) (see also Po´zniak, 1991). THEOREM 2.26. Let f ∈ C 2 (M) be a Morse function, bounded below, having only critical points of finite Morse index. Let g0 and g1 be complete Riemannian

48

A. ABBONDANDOLO AND P. MAJER

metrics on M, such that both ( f, g0 ) and ( f, g1 ) satisfy (PS) and the Morse – Smale property up to order 1. Then there is a chain complex isomorphism Φ: {C∗ ( f ), ∂∗ ( f, g0 )}
{C∗ ( f ), ∂∗ ( f, g1 )} of the form



Φx = x +

n(x, y)y,

∀x ∈ critk ( f ), k ∈ N,

(43)

y ∈ critk ( f ) f (y) < f (x)

for suitable integers n(x, y). The following lemma will be needed in the proof: LEMMA 2.27. Let a be a nondegenerate continuous symmetric bilinear form on the real Hilbert space (H, ·, ·), with either finite Morse index or finite Morse coindex. Let t0 ≥ 0, and let t → ·, ·t , t ∈ R, be a continuous path of inner products on H — equivalent to ·, · — constant for t ≥ t0 and for t ≤ −t0 . Let A(t) be the ·, ·t -self-adjoint bounded operator on H representing a with respect to the inner product ·, ·t : a(ξ, η) = A(t)ξ, ηt for every ξ, η ∈ H. Then the linear stable and unstable spaces of the path A (see Section 1.2) satisfy H = WAs ⊕ WAu . Proof. The path A is continuous and it is constant for t ≥ t0 and for t ≤ −t0 . Let us assume that a has finite Morse index, the other case being easily reducible to this one. The linear stable space WAs has dimension m(a), the Morse index of a, while the linear unstable space WAu is closed and has codimension m(a). Therefore, it is enough to prove that WAs ∩ WAu = (0). Let u0 ∈ WAs ∩ WAu , and let u: R → H be the solution of the linear Cauchy problem  u (t) = A(t)u(t), u(0) = u0 . Since A is constant for t ≥ t0 and for t ≤ −t0 , u(t) = e(t−t0 )A(t0 ) u(t0 ) ∀t ≥ t0 ,

u(t) = e(t+t0 )A(−t0 ) u(−t0 ) ∀t ≤ −t0 .

Since u(t) → 0 for |t| → 0, we deduce that u(t0 ) belongs to the negative eigenspace of A(t0 ), and u(−t0 ) belongs to the positive eigenspace of A(−t0 ). Since both A(t0 ) and A(−t0 ) represent the symmetric form a, we have

 a u(t0 ), u(t0 ) ≤ 0, a u(−t0 ), u(−t0 ) ≥ 0. (44) On the other hand, since A(t) represent a for every t, the inequality 

 1d
a u(t), u(t) = a u(t), u (t) = a u(t), A(t)u(t) = A(t)u(t), A(t)u(t)t ≥ 0 2 dt

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

49

is compatible with (44) if and only if u(t) = 0 for every t ∈ [−t0 , t0 ] (hence for every t ∈ R), proving that u0 = 0. Therefore WAs ∩ WAu = (0).
Proof of Theorem 2.26. We introduce the smooth Morse function ϕ: R → R,

ϕ(s) = 2s3 − 3s2 + 1,

which has two critical points, namely a local maximum at 0, with ϕ(0) = 1, and a local minimum at 1, with ϕ(1) = 0. Moreover ϕ (s) diverges for |s| → +∞.  = R × M consider the C 2 function On the manifold M  → R, f˜: M

f˜(s, p) = ϕ(s) + f (p).

It is a Morse function, with critical points of finite Morse index, and

 critk ( f˜) = {0} × critk−1 ( f ) ∪ {1} × critk ( f ) , for every k ∈ N. Therefore Ck ( f˜)
Ck−1 ( f ) ⊕ Ck ( f ),

∀k ∈ N,

(45)

the first group in the sum corresponding to the critical points in {0}× M, the second one to critical points in {1} × M. If χ: R → [0, 1] is a smooth cut-off function such that χ(s) = 1 for s ≤ 1/3  and χ(s) = 0 for s ≥ 23 , we can consider the complete Riemannian metric on M
 g˜ (s, p)[(σ, ξ), (σ , ξ )] = σσ + χ(s)g0 (p)[ξ, ξ ] + 1 − χ(s) g1 (p)[ξ, ξ ],  = R ⊕ T p M. for every (σ, ξ), (σ , ξ ) ∈ T (s,p) M
 Let (sn , pn ) be a (PS) sequence for ( f˜, g˜ ). Since ∇g˜ f˜(s, p)g˜ ≥ |ϕ (s)|, we can find a subsequence of (sn ) which converges either to 0 or to 1. Since ( f, g0 ) and ( f, g1 ) satisfy (PS) and g˜ (s, p)|(0)⊕T M is just g0 for s close to 0 and g1 for s close to 1, we conclude that ( f˜, g˜ ) satisfies (PS). Let us examine the negative gradient flow of f˜ with respect to the metric g˜ . (i) The hypersurfaces {0} × M and {1} × M are flow-invariant, and the restriction of the flow to {i} × M is nothing else but the negative gradient flow of f with respect to the metric gi , for i = 0, 1. Moreover the invariant set {0} × M is a repeller, while {1} × M is an attractor. Therefore: (ii) The only flow lines going from a critical point in {i} × M to a critical point in the same hypersurface are those which are fully contained in {i} × M, for i = 0, 1. (iii) There are no flow lines going from a critical point in {1} × M to a critical point in {0} × M.

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A. ABBONDANDOLO AND P. MAJER

 we have If we view f as a function on M, D f (s, p)[−∇g˜ f˜(s, p)] = D f (s, p)[(−ϕ (s), −∇g˜ (s,·) f (p)] = −∇g˜ (s,·) f (p)2g˜ (s,·) .

(46)

This implies that f is almost a Lyapunov function for the vector field −∇g˜ f˜: (iv) f decreases strictly on all the nonconstant orbits, apart from those of the form
 t → (s(t), x), with x ∈ crit( f ), s (t) = −ϕ s(t) . In particular, up to time shifts there is exactly one flow line going from (0, x) to (1, x), for x ∈ crit( f ), namely the orbit 
 s¯ (t) = −ϕ s¯(t) , t → ( s¯(t), x), with (47) s¯(0) = 12 .

 We claim that the intersection W u (0, x) ∩ W s (1, x) = ]0, 1[ × {x} is transverse. Indeed, by linearizing along the flow line ( s¯(t), x), we easily see that T (1/2,x) W u ((0, x)) = R ⊕ WAu ,

T (1/2,x) W s ((1, x)) = R ⊕ WAs ,

where the bounded linear operator A(t) : T x M → T x M is minus the Hessian of f at the critical point x with respect to the inner product g˜ ( s¯(t), x)|(0)⊕T x M = χ( s¯(t))g0 + (1 − χ( s¯(t)))g1 . Then A(t) represents the second differential of f at x with respect to the above inner product, so by Lemma 2.27, T x M = WAs ⊕ WAu . Therefore

  T (1/2,x) W u (0, x) ⊕ T (1/2,x) W s (1, x) = R ⊕ T x M = T (1/2,x) M, proving transversality. The vector field −∇g˜ f˜ need not satisfy the Morse – Smale condition up to order 1, but the only points where transversality can fail are the intersections of the unstable manifold of a critical point (0, x) with the stable manifold of a critical point (1, y), with x  y critical points of f . We can perturb the metric g˜ in order to achieve the Morse – Smale property up to order 1 without loosing the nice features (i) – (iv) of the vector field −∇g˜ f˜. More precisely, by Theorem 2.20, taking into  such that account (46), we can find a complete metric g on M (a) ( f˜, g) satisfies (PS); (b) g coincides with g˜ on the sets ]−∞, 13 ] × M, [ 23 , +∞[ × M, and R × U, where U ⊂ M is a neighborhood of crit( f ); (c) D f (s, p)[−∇g f˜(s, p)] < 0 if p  crit( f );

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51

(d) ( f˜, g) satisfies the Morse – Smale property up to order 1. Indeed, the function θ appearing in the statement of Theorem 2.20 can be chosen to vanish on the regions indicated in (b), where the intersections are already transverse, and to be so small that the metrics belonging to the space of perturbations satisfy (c). By property (b), the flow of −∇g f˜ still satisfies (i), (ii), (iii). By (c), it satisfies also (iv). We can now consider the Morse complex of ( f˜, g) relative to the sublevel ˜ { f < inf f − 1}. Notice that this sublevel contains no critical points. The boundary operator ∂k ( f˜, g) can be described by using Theorem 2.11 and Remark 2.13. To this purpose, it is convenient to choose the orientations of the unstable manifolds in the following way: since for every x ∈ crit( f ) there is a privileged isomorphism T x W u (x; −∇g0 f )
T x W u (x; −∇g1 f ), namely the restriction to the first space of the projection onto the first factor in the splitting T x M = T x W u (x; −∇g1 f ) ⊕ T x W s (x; −∇g1 f ), we can endow these two spaces with orientations which are compatible with this isomorphism. Then
 T (0,x) W u (0, x); −∇g f˜ = R ⊕ T x W u (x; −∇g0 f ) and


 T (1,x) W u (1, x); −∇g f˜ = (0) ⊕ T x W u (x; −∇g1 f )

can be given the product orientations by the the standard orientations of R and (0). In this way, we have chosen an orientation for the unstable manifold of each critical point of f˜. With this choice the transverse intersection

 W u (0, x) ∩ W s (1, x) = ]0, 1[ × {x} (48) is given the orientation corresponding to the vector ∂/∂s, which agrees with the direction of the flow. By (i), (ii), (iii), and (45) the boundary operator ∂k ( f˜, g): Ck−1 ( f ) ⊕ Ck ( f ) → Ck−2 ( f ) ⊕ Ck−1 ( f ) 

can be written as ∂k ( f˜, g) =

 ∂k−1 ( f, g0 ) 0 , Φk−1 ∂k ( f, g1 )

for some homomorphism Φk : Ck ( f ) → Ck ( f ). ˜ The fact that ∂∗ ( f , g) is a boundary, i.e., ∂k ( f˜, g) ∂k+1 ( f˜, g) = 0, implies that Φk−1 ∂k ( f, g0 ) = ∂k ( f, g1 )Φk ,

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that is (Φk )k∈N is a chain homomorphism from the Morse complex of ( f, g0 ) to the Morse complex of ( f, g1 ). By (iv), the intersection

  W u (0, x); −∇g f˜ ∩ W s (1, y); −∇g f˜ can be nonempty only if f (y) < f (x) or x = y, and in the latter case it consists of the single orbit (48). Together with the previous discussion on orientations, this fact implies that Φ has the form (43). So if we order the critical points of f with Morse index k by increasing value of f , we see that the homomorphism Φk is represented by an upper-triangular matrix, with 1 on the diagonal entries. A homomorphism of this form must be an isomorphism: this is well known when Ck ( f ) has finite rank, because in this case Φk is represented by a finite matrix with determinant 1, an invertible element of Z, but it remains true if the rank of Ck ( f ) is infinite. Indeed if x1 , x2 , . . ., are the critical points of index k ordered by increasing value of f , the inverse of Φk is defined inductively by Φ−1 k x1 = x1 ,

Φ−1 k xh = xh −

h−1 

n(xh , xi )Φ−1 k xi , ∀h ≥ 2.




i=1

EXERCISE 2.28. Generalize this result to the case of a strip {a < f < b}. EXERCISE 2.29. When f satisfies the condition (A8), it is possible to obtain the same conclusion of Theorem 2.26 by looking directly at the two cellular filtrations induced by the two negative gradient flows. Prove this fact. Then use the limit arguments of Section 2.9 to prove Theorem 2.26 under the hypothesis that ( f, g0 ) and ( f, g1 ) satisfy the Morse – Smale condition only up to order 0.

3. The Morse complex in the case of infinite Morse indices 3.1. THE PROGRAM

In this part we will consider a gradient-like C 1 vector field X on a Hilbert manifold M, whose rest points have infinite Morse index and co-index. In this case, the stable and the unstable manifolds of rest points are infinite-dimensional, and the flow of X does not produce a meaningful cellular filtration of M. Indeed, the infinite-dimensional Hilbert ball is retractable onto its boundary, so the rest points of X are homotopically invisible. However, we may hope that in some cases the unstable and the stable manifolds of pairs of rest points have finite-dimensional intersections. If this holds,

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we could use the formula for the boundary operator of Theorem 2.11 not as a description, but rather as the definition of the Morse complex. Our program is to follow this idea. Of course this program cannot be pursued in full generality. A first reason is that in general the unstable and stable manifolds may not have finite-dimensional intersections. A deeper reason is that the setting of gradient-like flows for a Morse function with critical points of infinite Morse index and co-index has too little rigidity. For instance, the following result was proved in Abbondandolo and Majer (2004). A sketch of the proof will be presented at the end of Section 3.3. THEOREM 3.1. Let f : M → R be a smooth Morse function on a separable Hilbert manifold, whose critical points have infinite Morse index and co-index. Let a: crit( f ) → Z be an arbitrary function. Then there exists a Riemannian metric g on M such that the corresponding negative gradient flow of f has the following property: for every pair of critical points x, y, the intersection W u (x) ∩ W s (y) is transverse and — if nonempty — has dimension a(x) − a(y). Moreover, the metric g can be chosen to be uniformly equivalent to any given metric g0 on M. Finally, if (xi , yi ), i = 1, . . . , k, are pairs of critical points such
 that xi and yi can be connected by a path ui : [0, 1] → M such that D f ui (t) [u i (t)] is negative for every t ∈ ]0, 1[, the metric g can be chosen in such a way that W u (xi ) ∩ W s (yi ) is not empty. Therefore the situation is drastically less rigid than the case of finite Morse indices, where the Morse index of a critical point does not involve the metric, and where we have seen that the isomorphism class of the Morse complex does not depend on the metric, and that its homology does not even depend on f . Let us examine another example of the lack of rigidity determined by infinite Morse indices and co-indices. We have seen that when the Morse indices are finite, the transverse intersection W u (x) ∩ W s (y) is always orientable, and each of its components has the same dimension m(x) − m(y). On the other hand, if Z is any separable Hilbert manifold (finite-dimensional or not, possibly with components of different dimension), there exists a smooth gradient-like flow on the Hilbert space H with exactly two rest points x and y, such that the intersection W u (x) ∩ W s (y) is transverse and diffeomorphic to Z × R (see Abbondandolo and Majer, 2003b, Section 4). These phenomena suggest that a Morse theory for functions f : M → R with critical points of infinite Morse index and co-index requires more structure than just the pair (M, f ). Our choice will be to consider a subbundle V of T M, suitably compatible with the gradient-like flow.

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3.2. FREDHOLM PAIRS AND COMPACT PERTURBATIONS OF LINEAR SUBSPACES

Before proceeding, we need to review some facts about the Hilbert Grassmannian Gr(H), the set of all closed linear subspaces of the separable Hilbert space H. See Abbondandolo and Majer (2003a) for a more complete presentation. If V ∈ Gr(H), we shall denote by PV the orthogonal projection onto V. The set Gr(H) is a complete metric space with the distance dist(V, W) := PV − PW . The connected components of Gr(H) are the subspaces Grn,m (H) = {V ∈ Gr(H) | dim V = n, codim V = m}, where n, m ∈ N ∪ {∞}, n + m = ∞. A pair (V, W) ∈ Gr(H)×Gr(H) is a Fredholm pair if V∩W is finite-dimensional and V + W is finite-codimensional. In this case, the number ind(V, W) := dim V ∩ W − codim(V + W) is said the Fredholm index of (V, W). The space of Fredholm pairs, denoted by Fp(H), is an open subset of Gr(H) × Gr(H), and the Fredholm index is a continuous (i.e., locally constant) function on it. See for instance Kato (1980, IV §4). Let W ∈ Gr(H). A closed linear subspace V is a compact perturbation of W if the operator PV − PW is compact. In this case, the pair (V, W ⊥ ) is Fredholm, and its index is said the relative dimension of V with respect to W, denoted by dim(V, W) := ind(V, W ⊥ ) = dim V ∩ W ⊥ − dim V ⊥ ∩ W. If (V, W) is a Fredholm pair and Z is a compact perturbation of V, then (Z, W) is still a Fredholm pair, and its index is ind(Z, W) = ind(V, W) + dim(Z, V).

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3.3. FINITE-DIMENSIONAL INTERSECTIONS

Let M be a smooth Hilbert manifold, and let X be a C 1 Morse vector field on M, with local flow φ: Ω(X) → M. We shall always assume that X has a Lyapunov function f . In view of Remark 1.21(ii), we shall assume that f ∈ C 2 (M) and that it is a nondegenerate Lyapunov function, meaning that for every x ∈
 rest(X) the quadratic form ξ → D2 f (x)[ξ, ξ] is coercive on E s ∇X(x) , while
 ξ → −D2 f (x)[ξ, ξ] is coercive on E u ∇X(x) . Let V be a smooth subbundle of T M, and let P be a projector onto V: P is a smooth section of the bundle of endomorphisms of T M such that for every p ∈ M, P(p) ∈ L(T p M) is a projector onto V(p). We shall assume the following compatibility conditions between X and V:

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 (C1) for every x ∈ rest(X), the positive eigenspace E u ∇X(x) of the Jacobian of X at x is a compact perturbation of V(x); (C2) for every p ∈ M, the operator (LX P)(p)P(p) is compact. Here LX P denotes the Lie derivative of P along X. By (C1), we can define the relative Morse index of the rest point x with respect to V to be the integer

  m(x, V) := dim E u ∇X(x) , V(x) . Condition (C2) depends only on the subbundle V, and not on the choice of the projector P onto it. Notice that the subbundle V is φ-invariant (in the sense that
 D2 φ(t, p)V(p) = V φ(t, p) for every (t, p) ∈ Ω(X)) if and only if (LX P)P = 0. Condition (C2) is equivalent to the fact that V is φ-essentially invariant:
 D2 φ(t, p)V(p) is a compact perturbation of V φ(t, p) , for every (t, p) ∈ Ω(X). When M is an open subset of the Hilbert space H, and V is a constant subbundle V ≡ V ∈ Gr(H), so that we can choose P ≡ PV , there holds (LX P)P = [DX, PV ]PV = (I − PV )DXPV .

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These assumptions have the following consequence. PROPOSITION 3.2. Assume that the Morse vector field X satisfies (C1) and (C2) with respect to the subbundle V. Then for every x ∈ rest(X): (i) for every p ∈ W u (x), T p W u (x) is a compact perturbation of V(p), with dim(T p W u (x), V(p)) = m(x, V); (ii) for every p ∈ W s (x), the pair (T p W s (x), V(p)) is Fredholm, with ind(T p W s (x), V(p)) = −m(x, V). So loosely speaking, W u (x) is essentially parallel to V, while W s (x) is essentially normal to V. Let us sketch the proof of the first claim in a simpler case: we assume that M is an open set of the Hilbert space H, and that V ≡ V ∈ Gr(H) is a constant subbundle. Let p ∈ W u (x), and let u(t) := φ(t, p) be the orbit of p. By linearization along u, using the notation of Section 1.2, we have that T p W u (x) = WAu ,

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 where A(t) := DX u(t) . By (C1), W := T x W u (x) = E u A(−∞) is a compact perturbation of V. By (C2), the operator [A(t), PV ]PV is compact for every t, and so is the operator [A(t), PW ]PW . Set B(t) := A(t) − [A(t), PW ]PW ,

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 so that B(−∞) = A(−∞) = DX(x), E u B(−∞) = W, and B(t)W ⊂ W for every t. These facts easily imply that WBu = W. On the other hand, since A(t) − B(t) is compact for every t, WAu is a compact perturbation of WBu = W, hence of V. By (51), T p W u (x) is a compact perturbation of V. The formula for its relative dimension with respect to V follows by continuity. The proof of claim (ii) is simpler. Since the set of Fredholm pairs is open and the index is locally constant, by (C1) the pair (T p W s (x), V(p)) is Fredholm of index −m(x, V) for every p ∈ W s (x) in a neighborhood of x. The tangent bundle T W s (x) is φ-invariant, and by (C2) the subbundle V is φ-essentially invariant, so these facts remain true for every p ∈ W s (x). By (49), Proposition 3.2 has the following easy corollary. COROLLARY 3.3. Assume that the Morse vector field X satisfies (C1) and (C2) with respect to the subbundle V. Let x, y ∈ rest(X), and assume that W u (x) and W s (y) meet transversally. Then W u (x) ∩ W s (y) is a submanifold of dimension m(x, V) − m(y, V). We conclude this section by sketching the proof of Theorem 3.1. By the already mentioned embedding theorem of Eells and Elworthy (1970), we can embed M as an open subset of the separable Hilbert space H. By modifying this embedding near the critical points of f , and by using the Morse Lemma (see for instance Palais, 1963), we may assume that f is quadratic near every critical point x: f (x + ξ) = f (x) + 12 A(x)ξ, ξ,

for |ξ| small,

for some self-adjoint invertible operator A(x). Fix a closed linear subspace V of H, with infinite dimension and codimension. By a further modification of the embedding, we may also rotate small neighborhoods of the critical points in such a way that the negative eigenspace E s (A(x)) of the operator A(x) is a compact perturbation of V, of relative dimension a(x). Here we actually need to use Kuiper’s theorem (Kuiper, 1965), stating the orthogonal group of H is contractible. It is now easy to build a vector field X having f as a nondegenerate Lyapunov function, and which satisfies (C1) and (C2) with respect to the constant subbundle V. Indeed, near a critical point x one may choose X to be the linear vector field X(x + ξ) = −∇ f (x + ξ) = −A(x)ξ,

for |ξ| small.

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Since the negative eigenspace of A(x) is a compact perturbation of V of relative dimension a(x), X satisfies (C1) and x has relative Morse index m(x, V) = a(x). The linear vector field X satisfies also (C2). Indeed by (50), (LX PV )PV = −(I − PV )A(x)PV = −PV ⊥ PE s (A(x)) A(x)PV − PV ⊥ A(x)PE u (A(x)) PV , and the operators PV ⊥ PE s (A(x)) and PE u (A(x)) PV are compact because E s (A(x)) is a compact perturbation of V. If p ∈ M is not a critical point, we may choose

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57

X to be the constant vector field X(p + ξ) = −∇ f (p), for every ξ so small that D f (p + ξ)[−∇ f (p)] < 0. Every constant vector field trivially satisfies (C2) with respect to the constant subbundle V. These local definitions of X can be patched together by a smooth partition of unity. In this way one can build a vector field X satisfying (52) near critical points, so that (C1) holds. The set of vector fields satisfying condition (C2) is a module over the ring of real functions, so X satisfies (C2). Having f as a Lyapunov function is a convex condition, so f is a Lyapunov function for X. Up to a small perturbation, we may assume that X also satisfies the Morse – Smale condition. Then Corollary 3.3 implies that W u (x) ∩ W s (y) is a submanifold of dimension m(x, V) − m(y, V) = a(x) − a(y). The fact that X is actually the negative gradient of f near the critical points makes it possible to find a metric g on M such that X = −∇g f . We refer to Abbondandolo and Majer (2004) for details on how to keep g uniformly equivalent to a given metric, and on how to obtain that W u (xi ) ∩ W s (yi ) is nonempty for every i = 1, . . . , k. 3.4. ESSENTIAL SUBBUNDLES

It is readily seen that if X satisfies (C1) and (C2) with respect to a subbundle V, then it satisfies (C1) and (C2) also with respect to a subbundle W which at every point is a compact perturbation of V. This fact suggests the possibility of weakening the structure, fixing only an essential subbundle of T M. In order to make this precise, we need to introduce the essential Grassmannians of a Hilbert space. See again Abbondandolo and Majer (2003b) for a complete discussion. The essential Grassmannian of H is the quotient of Gr(H) by the equivalence relation {(V, W) ∈ Gr(H) × Gr(H) | V is a compact perturbation of W}, and it is denoted by Gre (H). This space can also be seen as the space of symmetric projectors in the Calkin algebra L(H)/Lc (H) (Lc (H) denotes the closed ideal of compact operators). Notice that the finite-dimensional and the finite-codimensional spaces represent two points in Gre (H). We shall actually be interested in the complementary Gr∗e (H) of these two points, that is in the quotient of Gr∞,∞ (H). The (0)-essential Grassmannian Gr(0) (H) is the quotient of Gr(H) by the stronger equivalence relation {(V, W) ∈ Gr(H) × Gr(H) | V is a compact perturbation of W and dim(V, W) = 0}. Again, Gr∗(0) (H) denotes the quotient of Gr∞,∞ (H). The Bott periodicity theorem (see Bott, 1959), and the fact that the group of automorphisms of H which are

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compact perturbations of the identity is homotopy equivalent to the infinite general linear group GL(∞) = lim GL(n) (see Palais, 1965), allow to determine the −→ homotopy type of the essential Grassmannian, proving the following result. THEOREM 3.4. The quotient projection Gr∞,∞ (H) → Gr∗(0) (H) is a fiber bundle with contractible total space. The quotient projection Gr∗(0) (H) → Gr∗e (H) is a universal covering. The space Gr∗e (H) is path connected, its fundamental group is infinite cyclic, and if i ≥ 2,   Z if i ≡ 1, 5 mod 8,  
  Z2 if i ≡ 2, 3 mod 8, πi (Gr∗e (H))
πi−2 GL(∞) =    0 if i ≡ 0, 4, 6, 7 mod 8. Since the tangent bundle of an infinite-dimensional Hilbert manifold is always trivial (by the already mentioned Kuiper’s theorem; Kuiper, 1965), a subbundle V of T M can be identified with a map V: M → Gr(H). Similarly, an essential subbundle (respectively a (0)-essential subbundle) of T M can be identified with a map E: M → Gre (H) (resp. E: M → Gr(0) (H)). By Theorem 3.4, an essential subbundle E of T M can be lifted to a (0)essential subbundle if and only if the homomorphism 
E∗ : π1 (M) → π1 Gr∗e (H) = Z vanishes. A (0)-essential subbundle E of T M can be lifted to a true subbundle of T M if and only if all the homomorphisms
 E∗ : πi (M) → πi Gr∗(0) (H) vanish (a condition which has to be checked only for i ≡ 1, 2, 3, 5 mod 8). If the vector field X satisfies (C1) and (C2) with respect to a (0)-essential subbundle E of T M, then the relative Morse index m(x, E) can still be defined, and the conclusions of Proposition 3.2 and of Corollary 3.3 still hold (with the obvious changes). If the vector field X satisfies (C1) and (C2) with respect to an essential subbundle, there is no relative Morse index. In this case the transverse intersection W u (x) ∩ W s (y) is finite-dimensional, but different components may have different dimension. More precisely, the dimension of the connected component containing p depends on the homotopy class of the orbit of p, seen as a curve from (R, −∞, +∞) into (M, x, y). It is actually possible to construct an example of a gradient-like vector field on S 1 × H, which satisfies (C1) and (C2) with respect to a nonliftable essential subbundle, and has two critical points x and y such that the intersection W u (x)∩W s (y) is transverse and consists of two connected components of different dimension.

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3.5. ORIENTATIONS

We recall that in the case of finite Morse indices, an arbitrary choice of the orientation of all the unstable manifolds — or equivalently of the finite-dimensional spaces T x W u (x) — determines an orientation of each transverse intersection of unstable and stable manifolds. Now T x W u (x) is infinite-dimensional, so it does not carry orientations. The right object to orient turns out to be the Fredholm pair
 T x W s (x), V(x) . In order to deal with this question, we need to introduce the determinant bundle
 Det Fp(H) → Fp(H) on the space of Fredholm pairs (see Abbondandolo and Majer, 2003b, for more details). It is a real line bundle, whose fiber at (V, W) ∈ Fp(H) is 
∗ Det(V, W) := Det(V ∩ W) ⊗ Det H/(V + W) , where Det(Z) := Λdim Z (Z) denotes the space of top degree in the exterior algebra of the finite-dimensional vector space Z. Defining a bundle structure for this object is not immediate, because the maps (V, W) → V ∩ W and (V, W) → V + W are not continuous. We just mention the key ingredients in the constructions. The intersection map (V, W) → V ∩ W is continuous on the space of transverse pairs, while the sum (V, W) → V+W is continuous on the space of pairs with intersection (0). Then the bundle structure near a Fredholm pair (V0 , W0 ) can be constructed by fixing a finite-dimensional space Z such that Z + V0 + W0 = H and Z ∩ V0 = (0), and by replacing each pair (V, W) in a neighborhood of (V0 , W0 ) by (Z + V, W). Such a replacement turns out to be possible because of the existence of an exact sequence H 0 → V ∩ W → (Z + V) ∩ W → Z → → 0. V +W We recall that an exact sequence of finite-dimensional vector spaces 0 → Z1 → · · · → Zk → 0 induces a natural isomorphism " i odd

Det(Zi )


"

Det(Zi ).

i even

The space of Fredholm operators from H1 to H2 , denoted by F (H1 , H2 ) is “contained” in the space of Fredholm pairs of H1 × H2 . Indeed, the operator A ∈ L(H1 , H2 ) is Fredholm if and only if the pair (graph A, H1 × (0)) ∈ Gr(H1 × H2 ) × Gr(H1 × H2 ) is Fredholm, and the index is the same. The pullback of the determinant bundle on Fp(H) by the map
 F (H1 , H2 ) → Fp(H1 × H2 ), A → graph A, H1 × (0) ,

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A. ABBONDANDOLO AND P. MAJER

is the determinant bundle on the space of Fredholm operators, as defined by Quillen (1985). Let n ∈ N, and let
 Det Grn,∞ (H) → Grn,∞ (H) be the real line bundle whose fiber at Z ∈ Grn,∞ (H) is Det(Z). Let S be the set of
 all Z, (V, W) in  Grn,∞ (H) × Fp(H) n∈N

such that Z ∩ V = (0), and let Det(S) → S be the restriction to S of the tensor 

 product of the bundles n∈N Det Grn,∞ (H) and Det Fp(H) . The map
 S → Fp(H), Z, (V, W) → (Z + V, W), is continuous, and can be lifted to a continuous morphism between the corresponding determinant bundles:
 S : Det(S) → Det Fp(H) . The construction of such a morphism is based on the exact sequence 0 → V ∩ W → (Z + V) ∩ W →

H H Z+V
Z→ → → 0. V V +W Z+V +W

The morphism S is associative, meaning that if Z and Y are finite-dimensional linear subspaces of H such that Z ∩ Y = (Z + Y) ∩ V = (0), the diagram Det(Y) ⊗ Det(Z) ⊗ Det(V, W) S ⊗id



Det(Y + Z) ⊗ Det(V, W)

id⊗S

S

/ Det(Y) ⊗ Det(Z + V, W) 

S

/ Det(Y + Z + V, W)

commutes. An orientation of a finite-dimensional space Z can be defined as an orientation of the line Det(Z); similarly, an orientation of the Fredholm pair (V, W) is an orientation of the line Det(V, W). The morphism S allows to sum orientations: if
 Z, (V, W) ∈ S, the orientations of two objects among Z,

(V, W),

(Z + V, W),

determines an orientation of the other object. Let us go back to the question of orienting the intersections between unstable and stable manifolds. The assumption is that the vector field X satisfies (C1) and (C2) with respect to a subbundle V of T M. By assumption (C1), the pair

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(T x W s (x), V(x)) is Fredholm, for every x ∈ rest(X). Let us choose an orientation o(x) of such a Fredholm pair, for every rest point x, in an arbitrary way. Now let x, y be rest points such that W u (x) and W s (y) have transversal intersection. Let p ∈ W u (x) ∩ W s (y). By Proposition 3.2(ii), the pair (T p W s (y), V(p)) is Fredholm. Choose a closed complement V of T p (W u (x) ∩ W s (y)) in T p W s (y). By transversality, V is also a complement of T p W u (x) in T p M. It is a general fact in this case that the backward evolution of V with respect to the differential of the flow converges to T x W s (x): lim D2 φ(t, p)V = T x W s (x).

t→−∞


 Therefore, the Fredholm pair V, V(p) inherits by continuity an orientation
 from the orientation o(x) of T x W s (x), V(x) . On the other hand, the Fredholm
 pair T p W s (y), V(p) inherits an orientation from the orientation o(y) of
 T y W s (y), V(y) . The last two objects among

 T p W u (x) ∩ W s (y) , V, V(p) ,

u  
 T p W (x) ∩ W s (y) + V, V(p) = T p W s (y), V(p) are then oriented, so they induce an orientation of the first space. The construction continuously depends on p, hence it determines an orientation of W u (x) ∩ W s (y). We shall see in Section 3.7 that the orientations defined here satisfy a suitable coherence property. 3.6. COMPACTNESS

In the case of finite Morse indices, we have seen that the (PS) condition together with the positive completeness of X implies that W u (x) ∩ { f ≥ a} is precompact. Now the unstable manifold is infinite-dimensional, so this cannot be true, but we can hope W u (x) ∩ W s (y) to be precompact. However, assumptions (C1) and (C2) are not sufficient to get this result: W u (x) ∩ W s (y) may consist, for instance, of infinitely many flow lines going from x to y, with no cluster points besides x and y. We need to strengthen condition (C2), a local assumption, into a more global condition. We recall that that the Hausdorff distance of two subsets A, B of a complete metric space (W, d) is the number   distH (A, B) := max sup inf d(a, b), sup inf d(a, b) ∈ [0, +∞], a∈A b∈B

b∈B a∈A

and that the Hausdorff measure of noncompactness of A is the number βW (A) := inf{r > 0 | A can be covered by finitely many balls of radius r} ∈ [0, +∞],

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so that A is precompact if and only if βW (A) = 0. The function β is continuous with respect to the Hausdorff distance. Moreover, βW (A) coincides with the Hausdorff distance of A from the space of compact subsets of W: βW (A) = inf{distH (A, K) | K ⊂ W compact}.

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In the case of a normed vector space W, β has also the following properties: βW (λA) = |λ|βW (A), βW (A + B) ≤ βW (A) + βW (B),
 βW conv(A) = βW (A).

(54)

Let E be an essential subbundle of T M, different from the trivial essential subbundles [(0)] and [T M]. We shall assume that E admits a global presentation: there exists a smooth map Q : M → N into a Hilbert manifold such that for every p ∈ M, DQ(p) has finite-codimensional range, and ker DQ(p) belongs to the equivalence class E(p). For instance, E could be the equivalence class of a subbundle which is the vertical space of a submersion Q. We shall assume that N is endowed with a complete Riemannian metric, and we shall consider the induced metric on T N. The new assumption on the vector field X is: (C3)

(i) DQ ◦ X∞ < +∞; (ii) for every q ∈ N there exists δ = δ(q) > 0 and c = c(q) ≥ 0 such that 
 

βT N DQ X(A) ≤ c βN Q(A) ∀A ⊂ Q−1 Bδ (q) .

Let us restate this condition in a simple situation: assume that M is an open set of the Hilbert space H, and that E is the equivalence class of a constant subbundle V ∈ Gr(H). Then we can choose the global presentation to be the orthogonal projector onto W := V ⊥ , Q := PW . Denote by (XV , XW ) be the two components of X with respect to the orthogonal splitting H = V ⊕ W. Condition (C3)(i) says that XW is bounded, while (C3)(ii) is equivalent to: for every ξ ∈ W there exist δ > 0 and c ≥ 0 such that 

βW XW (A) ≤ c βW (PW A) ∀A ⊂ M ∩ V × (Bδ (ξ) ∩ W) . (55) In particular, if A ⊂ M is such that PW A is precompact, then also XW (A) = PW X(A) is required to be precompact. Thus, for every ξ ∈ M the map η → (I − PV )X(ξ + PV η) is a compact map in a neighborhood of 0. Therefore, the differential of this map at 0, namely (I − PV )DX(ξ)PV = (LX PV )(ξ)PV is compact. Hence (C3) implies (C2): the simple situation — M ⊂ H, E constant, Q projector — in which we have checked this fact is indeed the general local situation, and (C2) is a local assumption.

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Notice that in general (55) is strictly stronger than the fact that for every ξ ∈ W, XW should map (ξ + V) ∩ M into a precompact set, because (55) involves a Lipschitz control on the measure of noncompactness. However, these conditions are equivalent under a mild Lipschitz assumption on X. See Abbondandolo and Majer (2003b, Proposition 7.9) for a precise statement (in the case of a general map Q). The main result of this section is the following compactness theorem. THEOREM 3.5. Let E be an essential subbundle of T M with a global presentation Q: M → N into a complete Riemannian Hilbert manifold. Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3). Then for every pair of critical points x, y, the intersection W u (x) ∩ W s (y) is precompact. Let us sketch the proof. It is useful to introduce the following notion: a subset A ⊂ M is said essentially vertical if Q(A) is precompact. The proof is then based on the following steps: (i) if A is essentially vertical and t ≥ 0, then φ([0, t] × A) is essentially vertical; u (x) is essentially vertical; (ii) each local unstable manifold Wloc,r s (x) has precompact intersection with every (iii) each local stable manifold Wloc,r essentially vertical subset.

Let us prove (i) under the simplifying assumption that the target of the map Q is a Hilbert space E, and that the constants appearing in condition (C3)(ii) are uniform: c does not depend on q, and we can take δ = +∞. So (C3)(ii) becomes 

 βE DQ X(B) ≤ cβE Q(B) ∀B ⊂ M. (56)
 Let A ⊂ M be an essentially vertical set, that is βE Q(A) = 0. Since Q takes value in a Hilbert space, there holds 


 1 t DQ φ(s, p) X φ(s, p) ds, Q φ(t, p) = Q(p) + t · t 0 from which we deduce that 

  Q φ([0, t] × A) ⊂ Q(A) + [0, t] conv DQ X(φ([0, t] × A) . Then, by the properties (54) of the measure of noncompactness β and by (56),  


   βE Q φ([0, t] × A) ≤ βE Q(A) + t βE conv DQ X(φ([0, t] × A)  

  = t βE DQ X φ([0, t] × A) ≤ tc βE Q φ([0, t] × A) .

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 By the above inequality, βE Q φ([0, t] × A) vanishes for every t < 1/c, and by iteration, for every t ≥ 0. This proves (i). Since (ii) and (iii) are local statements, we may assume that the rest point x is the origin of the Hilbert space H, and that Q is the orthogonal projector with kernel V, a constant local representative of the essential subbundle E. By (C1),
 E u := E u ∇X(0) is a compact perturbation of V. This fact easily implies that a bounded set A ⊂ H is essentially vertical if and only if its projection Ps A on
 E s := E s ∇X(0) is precompact. In particular, the graph of a map σ: E u (r) → E s (r) is essentially vertical if and only if the map σ is compact. So (ii) can be restated by saying that the map σu : E u (r) → E s (r) whose graph is the local unstable manifold (see Theorem 1.12) is compact. By the graph transform method (see Shub, 1987, Chapter 5), σu is the fixed point of the contraction F, mapping every 1-Lipschitz

  map σ ∈ Lip1 E u (r), E s (r) into the map F(σ) ∈ Lip1 E u (r), E s (r) , whose graph is the φ-evolution at time 1 of the graph of σ, intersected with E u (r) × E s (r). So claim (i) implies that the contraction F maps the closed nonempty subspace of compact maps into itself, hence the fixed point σu is a compact map, proving (ii). s (x) is the graph of a Claim (iii) is an immediate consequence of the fact that Wloc,r continuous map σs : E s (r) → E u (r). Let us see how claims (i), (ii), and (iii) allow to conclude, in the case in which there are no rest points in the strip where f (y) < f (p) < f (x). Let (pn ) ⊂ W u (x) ∩ W s (y). We must prove that (pn ) has a converging subsequence. We can assume that x and y are not limit points of (pn ). Then we can find sn < 0 < tn such that u (x) ∩ { f = f (x) − }, φ(sn , pn ) ∈ Wloc,r

s φ(tn , pn ) ∈ Wloc,r (y) ∩ { f = f (y) + },

for some small  > 0. The fact that the are no rest points in the strip { f (y) < f < f (x)} implies that (tn − sn ) is bounded: otherwise by Remark 2.1, we could find


 a sequence rn ∈ [sn , tn ] such that D f φ(rn , pn ) X φ(rn , pn ) tends to zero, and
 by (PS) the sequence φ(rn , pn ) would have a subsequence converging to a rest point in the strip { f (y) +  ≤ f ≤ f (x) − }, a contradiction. By claim (ii), the set {φ(sn , pn ) | n ∈ N} is essentially vertical. By claim (i) and by the fact that (tn − sn ) is bounded, also the set {φ(tn , pn ) | n ∈ N} is essentially vertical. But the latter set is contained in the local stable manifold of y, so by claim (iii) it is precompact. Since (tn ) is bounded, also the sequence (pn ) is compact. In the general case, one needs the following stronger versions of (ii) and (iii): there exist arbitrarily small neighborhoods U of the rest point x such that if (pn ) converges to x then: (ii ) if tn ≥ 0 and φ(tn , pn )∂U then the set {φ(tn , pn ) | n ∈ N} is essentially vertical; (iii ) if sn ≤ 0 and φ(sn , pn )∂U then the set {φ(sn , pn ) | n ∈ N} has compact intersection with any essentially vertical subset.

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The proof of (ii ) and (iii ) makes use of Proposition 1.17. Then a combination of the argument shown above and the argument in the proof of Theorem 2.2(ii) allows to conclude the proof of Theorem 3.5.
REMARK 3.6. The requirement that the essential subbundle E should have a global presentation can be weakened, by replacing the map Q by a suitable family of maps Qi : Mi → Ni , i ∈ I, where {Mi }i∈I is an open covering of M. Besides allowing more general essential subbundles, this fact has also the advantage of localizing even more the constants appearing in assumption (C2)(ii).

3.7. TWO-DIMENSIONAL INTERSECTIONS

Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , and that (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3) with respect to a subbundle V of T M. In analogy with the finite indices case, we shall say that X satisfies the Morse – Smale property up to order k ∈ Z if W u (x) meets W s (y) transversally whenever m(x, V) − m(y, V) ≤ k. Let us study what happens when the Morse – Smale condition up to order 2 holds, and x, z are rest points with m(x, V) − m(z, V) = 2. Let W be a connected component of W u (x) ∩ W s (z). It is a two-dimensional manifold, and R acts freely on it. Therefore W/R is a connected one-dimensional manifold, that is it is either a circle or an interval. In the first case, it is easy to see that W = W ∪ {x, z} is a two-dimensional sphere, and the restriction of φ to W is topologically conjugated to the exponential flow on the Riemann sphere S 2 = C ∪ {∞}, R × S 2  (t, ζ) → et ζ ∈ S 2 . We shall be more interested in the second case, in which W is the union of W and two “broken orbits,” with exactly one intermediate rest point. More precisely, the situation is described by the following theorem. THEOREM 3.7. Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , and that (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3) with respect to a subbundle V of T M, and has the Morse – Smale property up to order 2. Let x, y be rest points with m(x, V) − m(z, V) = 2, and let W be a connected component of W u (x) ∩ W s (z) such that W/R is an interval. Then restriction of the flow φ to W is topologically conjugated to the product of two shift flows on R: there exists a continuous surjective map h: R × R → W with the following properties:

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(i) φ(t, h(u, v)) = h(u + t, v + t) for every (u, v) ∈ R × R, t ∈ R; (ii) h(R2 ) = W, and there exist rest points y, y with m(y, V) = m(y , V) = m(x, V) − 1, and W1 , W2 , W1 , W2 connected components of W u (x) ∩ W s (y), W u (y) ∩ W s (z), W u (x) ∩ W s (y ), W u (y ) ∩ W s (z), respectively, such that W1 ∪ W2  W1 ∪ W2 , and h(R × {−∞}) = W1 , h({−∞} × R) = W1 ,

h({+∞} × R) = W2 , h(R × {+∞}) = W2 ;

(iii) the restrictions of h to R2 , to {±∞}×R, and to R×{±∞}, are diffeomorphisms; (iv) denoting by deg the Z-topological degree, referred to the orientations defined in Section 3.5, there holds deg h = − deg h|{−∞}×R · deg h|R×{+∞} = deg h|R×{−∞} · deg h|{+∞}×R . When y  y , h is injective, so it is a conjugacy. When y = y , it may happen that W1 = W1 , or that W2 = W2 , but these identities cannot hold simultaneously. Statement (iv) expresses a form of coherence of the orientations defined in Section 3.5. Let us describe the main idea in the construction of h. By compactness and transversality, we can find a “broken orbit” in the closure of W, with exactly one intermediate rest point y of relative Morse index m(y, V) = m(x, V) − 1. Let W1 and W2 be the corresponding components of W u (x) ∩ W s (y) and W u (y) ∩ W s (z). Let p ∈ W1 , and let q ∈ W2 . Let A be a small hypersurface in W u (x) meeting W s (y) transversally at p, and let B be a small hypersurface in W s (z) meeting W u (y) transversally at q. Consider a neighborhood U of y of the form U = Eyu (r) × Eys (r), where r is so small that the local stable manifold of y is the graph of a θ-Lipschitz map σs : Eys (r) → Eyu (r), while the local unstable manifold of y is the graph of a θ-Lipschitz map σu : Eyu (r) → Eys (r), for some θ < 1. The forward evolution of A eventually intersects U in the graph of a θ-Lipschitz map from Eyu (r) to Eys (r): there is t0 ≥ 0 such that for every t ≥ t0 φ({t} × A) ∩ U = graph αt : Eyu (r) → Eys (r),

lip(αt ) ≤ θ,

and αt − σu ∞ → 0 for t → +∞. Similarly, for every t ≤ −t0 , φ({t} × B) ∩ U = graph βt : Eys (r) → Eyu (r),

lip(βt ) ≤ θ,

and βt − σs ∞ → 0 for t → −∞. Let u ≥ t0 and v ≤ −t0 . Since lip(αu ) ≤ θ < 1 and lip(βv ) ≤ θ < 1, the graphs of αu and of βv intersect in exactly one point, and we can define h(u, v) as h(u, v) := (graph αu ) ∩ (graph βv ).

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This defines h in a neighborhood of (+∞, −∞). See Abbondandolo and Majer (2003b, Section 11) for a complete proof. An analogous argument allows to prove the following result. PROPOSITION 3.8. Let x, y, z be rest points such that m(x, V) = m(y, V) + 1 = m(z, V) + 2, and let W1 , W2 be connected components of W u (x) ∩ W s (y), W u (y) ∩ W s (z), respectively. Then there exists a unique connected component W 

of W u (x)∩W s (z) such that W1 ∪ W2 belongs to the closure of φ(R × {p}) p ∈ W with respect to the Hausdorff distance.

3.8. THE MORSE COMPLEX

We now dispose of all the ingredients to build the Morse complex. The assumptions are that the Morse C 1 vector field X on the Hilbert manifold M is complete, satisfies (C1) – (C3) with respect to a subbundle V of T M, with a global presentation Q: M → N, that X satisfies the Morse – Smale condition up to order 2, has a nondegenerate Lyapunov function f ∈ C 2 (M), and that the pair (X, f ) satisfies (PS). For any k ∈ Z, denote by restk (X) the set of rest points x of X of relative Morse index m(x, V) = k, and let Ck (X) be the free Abelian group generated by restk (X). Assume the following finiteness condition: (C4) for every k ∈ Z, f is bounded below on restk (X). For every rest point x, we fix an orientation of the Fredholm pair (Tx W s (x),V(x)) in an arbitrary way. This choice induces an orientation of all the intersections W u (x) ∩ W s (y), for m(x, V) − m(y, V) ≤ 2. Let x, y be rest points of X with m(x, V) − m(y, V) = 1. Then W u (x) ∩ W s (y) is a 1-dimensional manifold with a free action of R, that is it is the union of the orbits of a discrete set of points. By Theorem 3.5 and by transversality, W u (x) ∩ W s (y) is compact: otherwise we could find a sequences of orbits in W u (x) ∩ W s (y) converging to a “broken orbit” from x to y, with at least one intermediate rest point, violating the Morse – Smale condition (up to order 0). Therefore, W u (x) ∩ W s (y) consists of finitely many orbits Wi , i = 1, . . . , h, each of which can be given a sign (Wi ) ∈ {+1, −1} depending on whether the direction of X agrees or does not agree with the orientation of Wi . In other words, if Wi = φ(R × {p}), (Wi ) is the degree of the map φ(·, p): R → Wi . We define the integer n(x, y) as n(x, y) =

h  i=1

(Wi ).

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By assumption (C4), we can define a homomorphism ∂k : Ck (X) → Ck−1 (X) generatorwise, as  n(x, y)y, ∀x ∈ restk (X). ∂k x = y∈restk−1 (X)

The results of Section 3.7 imply that these homomorphisms are boundary operators. PROPOSITION 3.9. For every k ∈ Z, ∂k−1 ∂k = 0. Proof. Let x and z be rest points with m(x, V) − m(z, V) = 2, and let S(x, z) be the set of “broken orbits” from x to z with exactly one intermediate rest point, necessarily of relative index m(z, V) + 1. By compactness and transversality, S(x, z) is a finite set. By Proposition 3.8, for every element W1 ∪ W2 of S(x, z) there is a unique connected component W of W u (x) ∩ W s (y) such that W1 ∪ W2 belongs to the closure of {φ(R × {p}) | p ∈ W} with respect to the Hausdorff distance. By Theorem 3.7, the closure of W contains exactly one other element W1 ∪ W2 , different from W1 ∪ W2 . So there is an involution W1 ∪ W2 → W1 ∪ W2 on S(x, z), without fixed points, and by Theorem 3.7 (iv), (W1 )(W2 ) = −(W1 )(W2 ).

(57)

If m(x, V) = k, the coefficient of z in ∂k−1 ∂k x is the number   n(x, y)n(y, z) = (W1 )(W2 ), y∈restk−1 (X)

which is zero by (57).

W1 ∪W2 ∈S(x,z)




Therefore, the Abelian groups Ck (X) and the homomorphisms ∂k , for k ∈ Z, are the data of a chain complex, called the Morse complex of X. The construction depends on the choice of the subbundle V, and on the choice of the orienta
 tions of T x W s (x), V(x) . Replacing the subbundle V by a compact perturbations produces a shift in the indices, equal to the relative dimension of the compact perturbation. A change of the orientations produces an isomorphic chain complex, the isomorphism being actually an involution. When the conditions (C1) – (C3) hold only with respect to a (0)-essential subbundle, there is no orientation theory available, and the above construction produces a chain complex of Z2 -vector spaces. Replacing the vector field X by another one (still satisfying conditions (C1) – (C3) with respect to the same subbundle V) having the same Lyapunov function f , produces an isomorphic Morse complex: the argument is analogous to the one used in the proof of Theorem 2.26. In particular, the homology of the Morse complex does not depend on the vector field, at it can be denoted by H∗ ( f ), the Morse homology of f .

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Although in this situation we should not expect the Morse homology H∗ ( f ) to be directly related to the singular homology of M, H∗ ( f ) is still considerably stable with respect to modifications of the function f . For instance, if (X0 , f0 ) and (X1 , f1 ) satisfy conditions (PS) and (C1) – (C4) (with respect to the same subbundle V), and if f1 − f0 is bounded, then the corresponding Morse homologies are isomorphic (see Abbondandolo and Majer, 2001, Theorem 1.8, but see also Theorem 1.10). This fact is a consequence of a more general functorial property of the Morse homology: Morse homology is a functor from the class of Morse functions which are Lyapunov functions of some vector field satisfying (PS) and (C1) – (C4), seen as a small category with the usual order relation, to the category of graded Abelian groups. In other words, to each inequality f0 ≥ f1 is associated a sequence of homomorphisms of Abelian groups φ f0 f1 : Hk ( f0 ) → Hk ( f1 ),

∀k ∈ Z,

such that φ f1 f2 φ f0 f1 = φ f0 f2 and φ f f = id. Actually, φθ◦ f f = id, if θ: R → R is a smooth function such that θ > 0 and θ(s) ≥ s. This fact is clearly useful in order to compute the Morse homology of a given function f : if one can squeeze f between two functions, f0 ≥ f ≥ f1 , the knowledge of the Morse homology of f0 and f1 and of the homomorphism φ f0 f1 allows to get information on the Morse homology of f . For instance, if φ f0 f1 is an isomorphism, then φ f0 f is injective and φ f f1 is surjective, hence the Morse homology of f is at least as rich as the Morse homology of f0 and f1 . The construction of the homomorphism φ f0 f1 involves the same idea used in the proof of Theorem 2.26: f0 and f1 can be used to build a new function f˜ on R × M, whose boundary operator ∂ is the cone of some homomorphism ψ f0 f1 from the Morse complex of f0 to the one of f1 . The ∂2 = 0 formula then implies that ψ f0 f1 is a chain map, so it induces a homomorphism φ f0 f1 in homology. Bibliographical note The Morse complex approach for compact manifolds When M is a compact manifold and X is the negative gradient flow of a smooth function, the relations (36) were proved by Morse (1925), see also Morse (1934; 1947). A classical reference for Morse theory is Milnor (1963). See also the review papers by Bott (1982; 1988). The dynamical system point of view arose after the seminal work of Smale, see Smale (1960; 1961) and the beautiful foundational paper Smale (1967), and it immediately had influences in topology, see Milnor’s book on the h-cobordism theorem (Milnor, 1965). In this framework, one can consider Morse – Smale flows, which are dynamical systems more general than gradient-like flows since they

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may have periodic orbits. The connection between Morse theory for Morse – Smale flows and the homotopy of the underlying manifold has been further clarified by Franks (1979), see also Franks (1980), and Cornea (2002a; 2002b). Interpreting the boundary homomorphism of a cellular filtration in terms of an algebraic count of the gradient flow lines connecting critical points of index difference 1, was already implicit in a paper by Thom (1949), who however did not clarify the conditions required on the gradient flow. This interpretation was pointed out by Witten (1982), where it is deduced quite indirectly from a relationship between Morse theory and certain deformations of the Laplace – Beltrami operator. The first explicit construction of the Morse complex is due to Floer (1989), see also Salamon (1990). Floer’s proof makes use of Conley index theory, a general and powerful method to decompose a dynamical system into simpler invariant sets, see Conley (1978), Conley and Zehnder (1984), and Salamon (1985). Weber (1993) contains a concise construction of the Morse complex, by dynamical systems techniques (see also Weber, 2005). A systematic study of the Morse complex of a function as a tool to build a homology theory which satisfies the Eilenberg – Steenrod axioms can be found in Schwarz (1993). Here the methods are closer to those used in Floer homology. The isomorphism with the singular homology is deduced by the fact that all the homology theories which satisfy the Eilenberg – Steenrod axioms are equivalent on compact CW-complexes. A more direct proof of this isomorphism, still in this spirit, can be obtained by interpreting singular homology theory in terms of pseudocycles, see Schwarz (1999). Banyaga and Hurtubise (2004) presents a self-contained exposition of Morse homology, adopting the dynamical system point of view and providing all the necessary tools from hyperbolic dynamics, as well as applications to Morse theory on Grassmannians and on Lie groups. The dynamical system point of view is at the basis of Harvey and Lawson’s approach to Morse theory in terms of the de Rham – Federer theory of currents (Harvey and Lawson, 2001). The idea is to construct a chain map from the complex of smooth differential forms to the complex of currents, by taking the limit for t → +∞ of the pullback of a differential form by the flow φ(t, ·). Such a chain map is chain homotopic to the inclusion, and it is a retraction onto the subcomplex of currents spanned by the stable manifolds of the flow. The cohomology of such a subcomplex is then isomorphic to the de Rham cohomology of the manifold, a result which implies the Morse relations (36). Infinite-dimensional Morse theory Morse theory for C 2 functions on Hilbert manifolds was developed by Palais (1963) and Smale (1964a; 1964b). The Palais – Smale condition was introduced in these papers. This version of Morse theory has been extensively used in the study of geodesics, see Klingenberg (1978; 1982). The first of these references contains also a description of the cellular complex approach to infinite-dimensional Morse

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theory, in the case of self-indexing functions. A complete presentation of infinitedimensional Morse theory including many applications to differential equations can be found in Mawhin and Willem (1989) and Chang (1993). Morse theory in the case of infinite Morse indices In simple situations, functions with critical points of infinite Morse index and co-index can be studied by taking finite-dimensional approximations. See, for instance, Chang, (1981; 1993), Conley and Zehnder, (1983; 1984), and Abbondandolo (2001). Another way of overcoming the lack of rigidity due to the presence of critical points of infinite Morse index and co-index is to restrict the class of admissible deformations to more rigid classes, as in Benci and Rabinowitz (1979) and Rabinowitz (1986). In the same spirit, a Morse theory for special classes of functions on a Hilbert space has been introduced by Szulkin (1992), and further refined by Abbondandolo (1997; 2000), Kryszewski and Szulkin (1997), Ge ba et al. (1999), and Izydorek (2001). The idea is to develop a generalized cohomology theory, which satisfies all the Eilenberg – Steenrod axioms except the dimension axiom. This axiom is replaced by the requirement that suitable infinite-dimensional spheres should have nontrivial cohomology. These generalized cohomologies will be functorial only with respect to restricted classes of continuous maps (the infinite-dimensional sphere is contractible), and it is possible to develop a Morse theory for functions whose gradient flow belongs to such a class. The idea of forgetting about the whole ambient space and looking only at the gradient flow lines connecting critical points is due to Floer, who applied it to a Cauchy – Riemann type equation which does not even produce a local flow (so the framework is quite different from the setting of these notes). See Floer (1988a; 1988b; 1988c; 1989), and the expository paper Salamon (1999). Angenent and van der Vorst (1999) have used this approach to study the gradient flow of a function associated to a class of elliptic systems. A complete study of the Morse complex approach in the case of functions on a Hilbert space consisting of a compact perturbation of a nondegenerate quadratic form has been carried on by the authors in Abbondandolo and Majer (2001). The results of Abbondandolo and Majer (2003b) summarized in the third part of these notes, allow a much more general setting. There is a large literature about the Hilbert Grassmannian, and related constructions. In particular, the space of all compact perturbations of an infinitedimensional and -codimensional closed linear subspace is called restricted Grassmannian by some authors (although sometimes this term is reserved for Hilbert-Schmidt perturbations). See for instance Sato (1981), Segal and Wilson (1985), Pressley and Segal (1986), Guest (1997), and Arbarello (2002). The role of these objects in the homotopy theory underlying Floer homology is discussed in Cohen et al. (1995).

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Nemirovski˘ı, A. S. and Semenov, S. M. (1973) The polynomial approximation of functions in Hilbert spaces, Mat. Sb. (N.S.) 21 (92), 255 – 277, Russian. Palais, R. S. (1963) Morse theory on Hilbert manifolds, Topology 2, 299 – 340. Palais, R. S. (1965) On the homotopy type of certain groups of operators, Topology 3, 271 – 279. Palis, J. (1968) On Morse – Smale dynamical systems, Topology 8, 385 – 405. Palis, Jr., J. and de Melo, W. (1982) Geometric Theory of Dynamical Systems, New York, Springer. Po´zniak, M. (1991) The Morse complex, Novikov cohomology and Fredholm theory, preprint, University of Warwick. Pressley, A. and Segal, G. (1986) Loop Groups, Oxford Math. Monogr., Oxford, Oxford Univ. Press. Quillen, D. (1985) Determinants of Cauchy – Riemann operators over a Riemann surface, Funct. Anal. Appl. 19, 31 – 34. Rabinowitz, P. H. (1986) Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc. Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1 – 41. Salamon, D. (1990) Morse theory, the Conley index and Floer homology, Bull. Amer. Math. Soc. 22, 113 – 140. Salamon, D. (1999) Lectures on Floer homology, In Y. Eliashberg and L. Traynor (eds.), Symplectic Geometry and Topology, Vol. 7 of IAS/Park City Math. Ser., Park City, UT, 1997, p. 143 – 229, Providence, RI, Amer. Math. Soc. Sato, M. (1981) Soliton equations as dynamical systems on an infinite-dimensional Grassman manifold, S¯urikaisekikenky¯usho K¯oky¯uroku 439, 30 – 46. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkh¨auser. Schwarz, M. (1999) Equivalence for Morse homology, In M. Barge and K. Kuperberg (eds.), Geometry and Topology in Dynamics, Vol. 246 of Contemp. Math., Winston-Salem, NC, 1998/San Antonio, TX, 1999, p. 197 – 216, Providence, RI, Amer. Math. Soc. ´ Segal, G. and Wilson, G. (1985) Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61, 5 – 65. Shub, M. (1987) Global Stability of Dynamical Systems, New York, Springer. Smale, S. (1960) Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66, 43 – 49. Smale, S. (1961) On gradient dynamical systems, Ann. of Math. (2) 74, 199 – 206. Smale, S. (1964)a A generalized Morse theory, Bull. Amer. Math. Soc. 70, 165 – 172. Smale, S. (1964)b Morse theory and a nonlinear generalization of the Dirichlet problem, Ann. of Math. (2) 80, 382 – 396. Smale, S. (1965) An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, 861 – 866. Smale, S. (1967) Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747 – 817. Szulkin, A. (1992) Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, 375 – 418. Thom, R. (1949) Sur une partition en cellules associ´ee a` une fonction sur une vari´et´e, C. R. Acad. Sci. Paris 228, 973 – 975. Weber, J. (1993) Der Morse – Witten Komplex, Master’s thesis, TU Berlin. Weber, J. (2005) The Morse – Witten complex via dynamical systems, Exposition. Math., to appear. Witten, E. (1982) Supersymmetry and Morse theory, J. Differential Geom. 17, 661 – 692.

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY ALBERTO ABBONDANDOLO Universit`a di Pisa MATTHIAS SCHWARZ∗ Universit¨at Leipzig

Abstract. Given the cotangent bundle T ∗ Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T ∗ Q which is fiberwise asymptotically quadratic, its well-defined Floer homology with the pair-of-pants ring structure is ring-isomorphic to the singular homology of the free loop space of Q endowed with its loop product. The analogous statement is true for the based loop space versions and the Pontrjagin product. This article gives an overview of the construction of this ring isomorphism which is based on Legendre duality and moduli spaces of flow trajectories of hybrid type, which are half Floer trajectories for the Hamiltonian problem and half Morse trajectories for the Lagrangian problem.

1. Introduction Initially, Floer homology had been invented in order to provide a proof of the nondegenerate Arnold conjecture for symplectic fixed points (Floer, 1988a, 1988b, 1989). Given a closed symplectic manifold (M 2n , ω) and a nondegenerate, timeperiodic Hamiltonian H: R/Z × M → R, the Arnold conjecture claims that there should be at least a minimal number of 1-periodic solutions x ∈ P1 (H) of the
 Hamiltonian differential equation x˙(t) = XH t, x(t) , that is, equivalently, fixed points of the time-1-map of the flow generated by the Hamiltonian vector field XH of H. This minimal number should be not smaller than the sum of the Betti numbers of M, that is rk H∗ (M). Floer’s key idea was to build formally a chain complex generated by the 1-periodic solutions P1 (H), a finite subset of the free loop space of M, and to use a Cauchy – Riemann type PDE for cylindrical maps u: R × R/Z → M connecting x, y ∈ P1 (H), in order to select and count particular trajectories in the free loop space in order to define a chain boundary operator. This Cauchy – Riemann type PDE makes use of Gromov’s theory of pseudoholomorphic curves (Gromov, 1985). The homology of the Floer chain complex turns out to be an invariant of the manifold M, independent of the Hamiltonian function H. In fact, this Floer homology HF∗ was proven to be isomorphic as a module over ∗

Work partially supported by DFG grant no. SCHW892/2-1.

75 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 75–108. © 2006 Springer. Printed in the Netherlands.

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some suitably chosen coefficient ring to the standard homology of M, or in view of Poincar´e duality, to the cohomology of M (e.g., Hofer and Salamon, 1995). Subsequently, further algebraic structures were constructed on this Floer homology, in particular a natural ring structure, the so-called pair-of-pants ring structure (Schwarz, 1995). In view of the above isomorphism to H ∗ (M), it was shown in Piunikhin et al. (1996) that, in general, the pair-of-pants ring structure is a deformation of the cohomology ring of M known as its quantum cohomology. However, all these results did not expose any nontrivial topological structure concerning the free loop space of M. This turns out to be essentially due to the compactness assumption on the symplectic manifold M. As soon as this is dropped and as also the Hamiltonian function H deviates essentially from being compactly supported, its associated Floer homology, if well-defined, tends to be a truly infinite-dimensional homology theory. In this paper, we focus on the case of cotangent bundles (M, ω) = (T ∗ Q, ωLiouv ) with their canonical symplectic structure and their canonical Lagrangian fibration by the cotangent spaces. A result first proven by Viterbo (1996) shows that Floer homology for the fiberwise quadratic Hamiltonian on T ∗ Q given by a Riemannian metric g on Q is isomorphic as a graded Abelian group to the singular homology of the free loop space of the manifold Q. Another proof of this result was given by Salamon and Weber (2003). Whereas Viterbo uses a method based on generating functions, Salamon and Weber use an approach relating the Cauchy – Riemann type PDE of Floer theory to the parabolic heat equation for the L2 -gradient of the energy functional on the free loop space of Q. The present article gives an outline of again another approach (Abbondandolo and Schwarz, 2004, 2005). Our aim was to find a more immediate construction on the level of the Floer chain complex which could also be used to analyze the pairof-pants ring structure of Floer homology, which clearly must correspond to some ring structure on the homology of the free loop space of the manifold Q. It is worth mentioning that here, in the context of noncompact symplectic manifolds T ∗ Q, the symmetry between a cohomology and a homology theory is finally broken. In fact, it turns out that the pair-of-pants ring structure of Floer homology corresponds to the loop product known from the work by Chas and Sullivan (1999).1 In our approach, the concrete tool to represent the homology of the free loop space of Q is Morse homology (Schwarz, 1993; Abbondandolo and Majer, 2005). The philosophy of our construction of the chain complex isomorphism between the Floer chain complex for H and the Morse chain complex of the free loop space of Q is to build so-called hybrid moduli spaces of connecting trajectories which are half Morse trajectories for the negative gradient flow of a Lagrangian action functional on the free loop space of Q, parametrized by “flow-time” from minus infinity 1 Here, we thank Octav Cornea for pointing out to us the presence of this preprint in November 2002 when already our first version of the ring isomorphism had been established.

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to zero, and half Floer trajectories in the free loop space of T ∗ Q parametrized from time zero to infinity. Coupling these two different types of half-trajectories is established via a Lagrangian boundary condition which uses crucially the fact that the Hamiltonian function on T ∗ Q on the Floer chain complex side is Legendre dual to the Lagrangian function on the tangent bundle T Q providing the Morse theory for the free loop space of Q. Counting such hybrid trajectories defines the chain complex homomorphism from the Morse chain complex into the Floer chain complex. It is intriguing to see that Legendre duality only allows such a construction in one direction, namely from the Lagrangian picture towards the Hamiltonian picture if using the negative gradient flow which corresponds to homology theories. Nevertheless, one can still show that this chain complex homomorphism has to be invertible. The result that Floer homology is isomorphic to the free loop space homology as a graded Abelian group already on the chain complex level has been proven for a large, fairly general class of Hamiltonian functions with fiberwise asymptotic quadratic upper and lower bound in Abbondandolo and Schwarz (2004). The proof that this isomorphism of homologies identifies the pair-of-pants product with the loop product of Chas-Sullivan is given in Abbondandolo and Schwarz (2005). Moreover, also the analogous versions for the based loop space are considered in both papers. Related results have been also achieved by Cohen (2005) and Ramirez (2005). In this article, we present a condensed outline of our results from Abbondandolo and Schwarz (2004; 2005). The main result is given in Theorem 2.7 below. The present article is based on a series of talks given by the second author during the workshop SMS 2004 in Montr´eal. Both authors wish to thank the organizers Paul Biran, Octav Cornea and Franc¸ois Lalonde for the very stimulating atmosphere of this workshop. We are also indebted to Ralph Cohen, Octav Cornea, Yong-Geun Oh, Dietmar Salamon, Dennis Sullivan, Claude Viterbo and Joa Weber for many fruitful discussions. 2. Main result 2.1. LOOP SPACE HOMOLOGY

Let us briefly recollect the classical and well-known facts about loop space homology which will be extended by the loop product ring structure in the setting of manifolds. Let (Q, q0 ) be a pointed topological space and denote its free and based loop space respectively by ΛQ = C 0 (T, Q), T = R/Z, ΩQ = {c ∈ ΛQ | c(0) = q0 }.

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We have the classical Serre fibration i

ev

ΩQ → ΛQ −→ Q,

ev(c) = c(0).

Without further remarks we will denote in the following H∗ for singular homology with integer coefficients. We have the continuous concatenation injection #: ΩQ × ΩQ → ΩQ, (c, c ) → c # c ,  c(2t), 0 ≤ t ≤ 12 , c # c (t) = c (2t − 1), 12 ≤ t ≤ 1,

(1)

which induces the homomorphism #∗ : H∗ (ΩQ) ⊗ H∗ (ΩQ) → H∗ (ΩQ)

(2)

of graded groups and for which we have the classical result in the case of coefficients in a field k (see for instance F´elix et al., 2001), THEOREM 2.1. (H∗ (ΩQ; k), #∗ ) is a graded Hopf algebra. From now on, let us assume that Q is a smooth, oriented, n-dimensional manifold. Moreover, for simplicity we will restrict our perspective throughout this article to compact manifolds without boundary. For analytical reasons we restrict ΛQ and ΩQ to dense subspaces which carry the structure of smooth Hilbert manifolds. In fact, given any smooth embedding Q ⊂ RN , the Hilbert space W 1,2 (T, RN ) of square integrable functions with weak derivatives in L2 defines a smooth structure of a Hilbert manifold on W 1,2 (T, Q) = {c ∈ W 1,2 (T, RN ) | c(T) ⊂ Q}. Using an auxiliary Riemannian structure g on Q, we obtain a well-defined Riemannian structure on W 1,2 (T, Q) as a Hilbert manifold by 

  gc(t) v(t), w(t) + gc(t) (∇c˙ v, ∇c˙ w) dt. v, w = T

Clearly, ev−1 (q0 ) = {c ∈ W 1,2 (T, Q) | c(0) = q0 } is a smooth submanifold of codimension n and it is well-known that the dense inclusions W 1,2 (T, Q) → C 0 (T, Q) = ΛQ, ev−1 (q0 ) → ΩQ, are homotopy equivalences. We now use the notations ΛQ and ΩQ for the smooth Hilbert manifolds. Summing up, we observe

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LEMMA 2.2. ΩQ → ΛQ is a smooth embedding of codimension n with oriented ev normal bundle ν → ΩQ, and ΩQ → ΛQ −→ Q is a locally trivial fiber bundle. Proof. The first assertion follows from the fact that the map ev is a submersion   onto an n-dimensional oriented manifold. Find open coverings Q = Uα = Vα , Uα ⊂⊂ Vα , with marked points xα ∈ Uα and smooth maps hα : Uα → Diff Q in the sense that hα : Uα × Q → Q is smooth and hα (x, ·) is a diffeomorphism for all x ∈ Uα , with hα (x, xα ) = x and hα (x, Uα ) ⊂ Vα ∀ x ∈ Uα , hα (x, y) = y ∀x ∈ Q \ Vα . Then consider ΛQ|Uα = ev−1 (Uα ) = {c ∈ ΛQ | c(0) ∈ Uα } and define φα : ΛQ|Uα → ev−1 (xα ) × Uα ,

  φα (c) = hα c(0) −1 ◦ c, c(0) . We have the fiber diffeomorphism −1 −1 φ−1 α (·, x): ev (xα ) → ev (x)

for all x ∈ Uα , and ev: ΛQ → Q is a locally trivial fiber bundle.




Note that the normal bundle ν of ΩQ = ev−1 (q0 ) can be identified with |ΩQ . Since ν is an oriented rank-n vector bundle, we have a well-defined Thom class u ∈ H n (ν(ΩQ), ν(ΩQ) \ OΩQ ) ev∗ T Q

such that the Thom isomorphism


∩ u: H∗ (ν, ν \ O) − → H∗−n (ΩQ) yields the definition of ∩: H∗ (ΛQ) → H∗−n (ΩQ)

(3)

by composing j∗







H∗ (ΛQ) −→ H∗ (ΛQ, ΛQ \ ΩQ) − → H∗ (ν, ν \ O) − → H∗−n (ΩQ), where j∗ is taken from the long exact sequence and the middle isomorphism is excision. Note that this definition of ∩: H∗ (ΛQ) → H∗−n (ΩQ) uses the manifold property of Q and its orientability in an essential way. Let us now consider in an analogous manner the codimension-n-submanifold X = {(c, d) ∈ ΛQ × ΛQ | c(0) = d(0)},

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consisting of figure-eight-loops. Similarly, the normal bundle is oriented and we can define ∩: H∗ (ΛQ × ΛQ) → H∗−n (X). (4) This allows now the following DEFINITION 2.3. The loop product o: H∗ (ΛQ) ⊗ H∗ (ΛQ) → H∗−n (ΛQ)

is given by the composition ×

∩

#∗

→ H∗ (ΛQ × ΛQ) − → H∗−n (X) −→ H∗−n (ΛQ). H∗ (ΛQ) ⊗ H∗ (ΛQ) − In view of the intersection product •: H∗ (Q) ⊗ H∗ (Q) → H∗−n (Q) which is well-defined for the oriented manifold Q, we have PROPOSITION 2.4. The following are ring homomorphisms, ∩: (H∗ (ΛQ), o) → (H∗−n (ΩQ), #∗ ), ev∗ : (H∗ (ΛQ), o) → (H∗ (Q), •). Our aim is to find precisely this triple of homological ring structures o, #∗ and • with ring homomorphisms isomorphically within the framework of Floer homology. We mention here again, that the loop product was already constructed in Chas and Sullivan (1999), considered in a similar context in Cohen and Jones (2002) and Cohen and Voronov (2005), and that similar results relating the ring structures to Floer theory are also contained in Cohen (2005) and Ramirez (2005). 2.2. FLOER HOMOLOGY FOR THE COTANGENT BUNDLE

Here we recall briefly the general setup of Floer homology for Hamiltonian systems before we apply it to the situation of cotangent bundles. Let (P2n , ω) be a symplectic manifold, H: T × P → R a smooth Hamiltonian, explicitly time-dependent and 1-periodic in time t. Let XH be the associated nonautonomous vector field defined by ω(XH , ·) = −dH and denote the set of 1-periodic solutions by
 PΛ (H) = {x ∈ ΛP | x˙(t) = XH t, x(t) ∀ t ∈ T}. Assume that H is nondegenerate, i.e., we have transverse intersection graph φ1H  P ⊂ P × P for the time-1-map of the nonautonomous flow φtH of XH , see also

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(H0 )Λ below in Section 4. Let us also assume that we have an integral grading µΛ : PΛ (H) → Z by the Conley – Zehnder index. Given a t-dependent almost complex structure J: T × P → End(T P) on P, compatible with ω (i.e., ω(·, J(t)·) Riemannian metric ∀ t ∈ T) we define for x, y ∈ PΛ (H) the space of parametrized Floer trajectories

M x,y (J, H) = u: R × T → P lim u(s, ·) = x, lim u(s, ·) = y, s→−∞ s→+∞ 
 u s − J(t, u) ut − XH (t, u) = 0 . Througout this article we make frequent use in an ad hoc manner of the symbol M denoting varying kinds of solution spaces. Note the choice of the sign in the (anti-)CR-operator which implies that u ∈ M x,y can be viewed as a negative flow line for the unregularized L2 -gradient of the Hamiltonian action functional. The first essential compactness result of Floer theory states that for generic #x,y = M x,y (J, H)$R of unparametrized trajectories divided by the J, the space M R-shift-action, for x  y, is a manifold of dimension µΛ (x) − µΛ (y) − 1, and that it is compact in dimension 0 and compact up to simple splitting of trajectories in dimension 1, provided that (a) P is a closed symplectic manifold and the cohomology classes {ω}, c1 (T M, J) ∈ H 2 (M) satisfy suitable conditions with respect to π2 (M), or (b) P is noncompact, {ω}, c1 satisfy suitable conditions and J, H have a suitable asymptotic behaviour on the end of P. Below, the latter case will be made precise for cotangent bundles P = T ∗ Q. Next, we set for x, y ∈ PΛ (H)  # , µΛ (x) − µΛ (y) = 1, # M x, y = alg x,y 0, else, where #alg denotes counting of trajectories with signs according to their coherent orientation (Floer and Hofer, 1993). We define the finitely generated, graded, free Abelian group CFΛ ∗ (H) = Λ P (H) ⊗ Z and the boundary operator Λ ∂: CFΛ ∗ (H) → CF∗−1 (H),   ax x = a x x, yy. ∂ x

µΛ (y)=µΛ (x)−1

Note that this is the simplest possible version of the Floer chain complex without the use of Novikov rings (see Hofer and Salamon, 1995), which will be sufficient for our framework of cotangent bundles. The main theorem of Floer theory is

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THEOREM 2.5 (Floer et al.).
 (a) CF∗ (H), ∂ = ∂(J, H) is a chain complex, i.e., ∂2 = 0 and thus HF∗ (J, H) = ker ∂/ im ∂ is well-defined. (b) If P is a closed symplectic manifold (with suitable conditions on ω|π2 and c1|π2 ), then HF∗ (H, J)
H n−∗ (P) are isomorphic in a natural way. Note the shift in the grading for the isomorphism of Floer homology with singular homology of P. This is due to the chosen normalizations for the Conley – Zehnder index. The choices are such that the grading µΛ on PΛ (H) and the Hamiltonian proportional on iteration action functional (see below) are positively   (m) (m) Λ (−1) Λ = −µ (x) under time-reversal , x (t) = x(mt) and µ x towers x m∈Z which in view of (b) corresponds to Poincar´e duality H n−∗ (P)
Hn+∗ (P). Let us now consider this concept of Floer homology for cotangent bundles of smooth n-dimensional manifolds Q which we assume (mainly for reasons of simplicity) to be closed and oriented. P = T ∗ Q carries the canonical symplectic structure ω = dθLiouv where the Liouville-1-form is canonically given by θLiouv =

n 

pi dqi

i=1

in local coordinates (q1 , . . . , qn ) ∈ Q. Let again H: T × T ∗ Q → R be a nondegenerate Hamiltonian and consider x, y ∈ PΛ (H) and MΛx,y = M x,y (J, H) as above. Note that, for the exact symplectic form ω = dθLiouv we have the Hamiltonian action functional for paths or closed curves x in T ∗ Q defined by  1
 
∗ AH (x) = x θLiouv − H t, x(t) dt . 0

There is also a version of Floer homology for the case of the based loop space ΩQ. Consider q0 ∈ Q and the Lagrangian submanifold T q∗0 Q ⊂ (T ∗ Q, ω). Then, let 


 x, y ∈ PΩ (H) = x: [0, 1] → T ∗ Q x˙(t) = XH t, x(t) , x(0), x(1) ∈ T q∗0 Q ,


 ∗ MΩ x,y (J, H) = u: R × [0, 1] → T Q u s = J(t, u) ut − XH (t, u) , u(s, 0), u(s, 1) ∈ T q∗0 Q ∀ s ∈ R,  lim u(s, ·) = x, lim u(s, ·) = y . s→−∞

s→+∞

Here, we have to assume H to be nondegenerate with respect to q0 , that is for x ∈
 PΩ (H) the linearized time-1-map Dφ1H x(0) maps T q∗0 Q = T x(0) (T q∗0 Q) transverse

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83

to T q∗0 Q = T x(1) (T q∗0 Q). We also have a Z-grading by the Maslov index µΩ for paths of Lagrangian subspaces, see (H0 )Ω below in Section 4. We now try to define Λ resp. Ω

∂: CF∗

Λ resp. Ω

(J, H) → CF∗−1

(J, H)

for nondegenerate H and generic J. The crucial new problem is to guarantee the #x,y of unparametrized trajectonecessary compactness properties of the spaces M ries. Since ω = dθ is exact, there is no obstruction by bubbling-off, but we need suitable asymptotic conditions on H and J in order to grant a priori C 0 -estimates for trajectories u ∈ M x,y (J, H). Pick any Riemannian metric g on Q. The associated Levi-Civita connection provides a canonical ω-compatible almost complex structure JLC by mapping vertical tangent spaces onto horizontal ones and reversely. The main assumption on the admissible class of Hamiltonian functions H: T × T ∗ Q → R is the following:
 For any coordinate chart U, (q1 , . . . , qn ) on Q (H1 ) there exist constants h0 > 0 and h1 ≥ 0 such that % ∂& dH(t, q, p) p − H(t, q, p) ≥ h0 |p|2 − h1 , ∂p (H2 ) there exists a constant h2 ≥ 0 such that |∂q H(t, q, p)| ≤ h2 (1 + |p|2 ),

|∂ p H(t, q, p)| ≤ h2 (1 + |p|),

for all t ∈ T, p ∈ T q∗ Q, q ∈ U. Here, p∂/∂p is the canonical Liouville vector field, ω(p∂/∂p, ·) = θLiouv . (H1 ) essentially states that H grows at least fiberwise quadratically like 12 h0 |p|2 − const and (H2 ) implies that H grows at most fiberwise quadratically like 12 h2 |p|2 + const, and it is radially convex in p for |p| large. Note that apart from the size of the concrete constants h0 , h1 , h2 , the form of the estimates (H1 ) and (H2 ) is invariant under coordinate changes on Q and independent of the choice of g. A typical example is given by physical Hamiltonians such as H(t, q, p) = 12 |p − A(t, q)|2 + V(t, q). The asymptotic condition on the ω-compatible almost complex structure J is that for a suitable constant j0 depending on g (J) J − JLC ∞ < j0 . The first result now states that we have well-defined Floer homology for the given class of Hamiltonian functions on the cotangent bundle.

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THEOREM 2.6 (Abbondandolo and Schwarz, 2004). Given H and a generic J on P = T ∗ Q satisfying (H1 ), (H2 ), (J) above, the Floer chain complexes 

Λ CF∗ (H), ∂(J, H) and CFΩ ∗ (H), ∂(J, H) are well-defined and for any generic J, Ω J satisfying (J) there exist quasi-isomorphisms providing HFΛ ∗ (H) and HF∗ (H) independent from J. Moreover, given two Hamiltonians H, H satisfying (H1 ),

Ω (H2 ) there exist natural isomorphisms HFΛ → HFΛ → ∗ (H) − ∗ (H ) and HF∗ (H) − HFΩ ∗ (H ) satisfying the cocycle condition. The latter isomorphisms are called the continuation isomorphisms. The central result of the present article formulates the full context of the ring isomorphisms of the above Floer homology with loop space homology. THEOREM 2.7 (Abbondandolo and Schwarz, 2005). Given a Hamiltonian H on the cotangent bundle T ∗ Q satisfying above conditions (H1 ) and (H2 ), we have the following additional structures and relations for its Floer homology. Ω (a) HFΛ ∗ and HF∗ carry natural bilinear operations       (r) (s) (r+s) ⊗ HFΩ → HFΩ , mΩ : HFΩ k H l H k+l H       (r) (s) (r+s) ⊗ HFΛ → HFΛ , mΛ : HFΛ k H l H k+l−n H

for all k, l ∈ Z, r, s ∈ N with H (r) (t, ·) = rH(rt, ·). (b) There exist natural homomorphisms compatible with mΩ and mΛ ,

Λ Ω → HFΩ h: HFΛ ∗ (H), m ∗−n (H), m ,
Λ e: HFΛ → (H∗ (Q), •). ∗ (H), m (c) There exist natural Z-module isomorphisms compatible with the ring structures mΩ and o, respectively mΛ and #∗ ,
Λ
− → (H∗ (ΛQ), o), ΦΛ : HFΛ ∗ (H), m Ω
Ω Ω
Φ : HF∗ (H), m − → (H∗ (ΩQ), #∗ ) such that the diagram (H∗ (Q), •) o

e


Λ Λ HF∗ , m

h

ΦΛ


(H∗ (Q), •) o commutes.

ev∗



(H∗ (ΛQ), o)

/
HFΩ , mΩ  ∗−n ΦΩ


∩

 / (H∗−n (ΩQ), #∗ )

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85

The isomorphisms ΦΛ and ΦΩ as Z-module-isomorphisms are already given in Abbondandolo and Schwarz (2004). All statements about the ring structures are proven in Abbondandolo and Schwarz (2005). The following remarks should be made: (i) By naturality in (a) – (c) we mean two aspects: One is the compatibility with
the canonical continuation isomorphism HF∗Λ/Ω (H) − → HF∗Λ/Ω (H ) associated to changes of the Hamiltonians H, H both satisfying (H1 ) and (H2 ). The other important aspect is that these homological structures are already defined on chain level. (ii) The homomorphisms e and ev∗ are endowed with natural right inverses 
Λ i: (H∗ (Q), • → HFΛ ∗ (H), m ,
 i∗ : H∗ (Q), • → (H∗ (ΛQ), o), i.e., e ◦ i = idH∗ (Q) = ev∗ ◦i∗ , where i∗ is induced by the inclusion of constant loops, Q → ΛQ, c(t) = q, t ∈ T. The latter right-inverse property is seen in an analogous manner as Proposition 2.4 by standard algebraic topology. (iii) Not explicitly mentioned above is a natural R-filtration on HFΛ , HFΩ by the values of the Hamiltonian action functional. The “ring structures” mΛ , mΩ can be shown to be fully compatible with that filtration. (See (9) below.) (iv) It is an intriguing question, open to us, whether the Hopf algebra structure Ω Ω of (HFΩ ∗ , m ) implied by the isomorphism Φ can be seen immediately on the side of Floer homology. 3. Ring structures and ring-homomorphisms In this section we sketch the definitions of mΛ , mΩ , h, e, i and ΦΛ resp. Ω . Full details will be contained in Abbondandolo and Schwarz (2005). 3.1. THE PAIR-OF-PANTS PRODUCT

3.1.1. Based loop space From now on, the symbol # will denote concatenation in time without rescaling. We first construct mΩ for the based loop space case. Let Σ = R × [0, 1] ∪˙ R × [1, 2]/(s, 1−) ∼ (s, 1+) for s ≥ 0 i.e., the interior Σ˚ = R×(0, 2)\(−∞, 0)×{1} carries the standard complex structure as an open subset of C, and Σ is a noncompact Riemann surface with boundary √ where local conformal boundary coordinates at (0, 1) ∈ Σ are given by z → z ∈

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A. ABBONDANDOLO AND M. SCHWARZ

{z ≥ 0}. For s ≤ 0 we distinguish the different boundary points of Σ by the notation (s, 1+) and (s, 1−). Let H: T × T ∗ Q → R and q0 ∈ Q be such that both

∗ ∗ PΩ 1 (H) = {x: [0, 1] → T Q | x(0), x(1) ∈ T q0 Q, x˙ = XH (t, x) , ∗ ∗ PΩ 2 (H) = {x: [0, 2] → T Q | x(0), x(2) ∈ T q0 Q, x˙ = XH (t, x)}

consist only of nondegenerate solutions. Observe that we have the canonical identification P2 (H) = P1 (H (2) ), H (2) (t, x) = 2H(2t, x) by rescaling. Moreover, we add the further, generic nondegeneracy assumption Ω Ω that x(1)  T q∗0 Q for all x ∈ PΩ 2 (H). Then, given x, y ∈ P1 (H) and z ∈ P2 (H), we define the solution set  Ω ∗ ∞ ˚ 0 M x,y;z = u: Σ → T Q ∈ C (Σ) ∩ C (Σ)
 ˚ u s − J(t, u) ut − XH (t, u) = 0, ∀ (s, t) ∈ Σ,  x(t), 0 ≤ t ≤ 1−, lim u(s, t) = y(t − 1), 1+ ≤ t ≤ 2, s→−∞ lim u(s, t) = z(t), 0 ≤ t ≤ 2,  s→∞ u(s, t) ∈ T q∗ Q, ∀ [s, t] ∈ ∂Σ = Σ \ Σ˚ . 0

Note that J in general has to be allowed to be dependent on t ∈ [0, 2] and J(t + 1, ·)  J(t, ·), 0 ≤ t ≤ 1.
 PROPOSITION 3.1. For a generic choice of J(t, ·) t∈[0,2] and if x # y  z, the Ω Ω Ω solution set MΩ x,y;z is a manifold of dimension µ (x) + µ (y) − µ (z). An important feature is the sharp energy estimate  ∞ 2 |u s |2J ds dt = AH (x) + AH (y) − AH (z) ≥ 0. −∞

(5)

0

To allow for more general situations, we consider nondegenerate H: [0, 1] × T ∗ Q → R and K: [1, 2] × T ∗ Q → R with H(1, ·) = K(1, ·) and we let H # K: [0, 2] ×

Figure 1.

Σ, strip with a slit

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T ∗ Q be the concatenation which generically is again nondegenerate. Then, more generally, we will construct the operation Ω Ω mΩ : HFΩ ∗ (H) ⊗ HF∗ (K) → HF∗ (H # K),

(6)

where H # K is again viewed as rescaled over [0, 1] like above. By compatibility with the continuation isomorphisms we obtain also the operation mΩ for K = H and H # K = H (2) . PROPOSITION 3.2. Under the assumptions (H1 ) and (H2 ) for H, K and generic J satisfying (J) we have: (a) MΩ x,y;z is compact in dimension 0, (b) in dimension 1, MΩ x,y;z is compact up to simple splitting of Floer trajectories of relative index 1 at at most one end of Σ. As a consequence, we can define the operation mΩ on chain level Ω Ω mΩ : CFΩ k (H) ⊗ CFl (K) → CFk+l (H # K),  #alg MΩ mΩ (x, y) = x,y;z (H # K, J) z,

(7)

µ(z)=µ(x)+µ(y)

and it satisfies ∂H#K ◦ mΩ (x, y) = mΩ (∂H x, y) ± mΩ (x, ∂K y).

(8)

Due to the energy estimate (5), mΩ is compatible with the R-filtration by the Hamiltonian action,    a x x ∈ CF∗ (H) a x = 0 for AH (x) > a , (9) CFa∗ (H) = x∈P(H)

Ω

m : CF (H) ⊗ CFb (K) → CFa+b (H # K). a

3.1.2. Free loop space The construction of mΛ for HFΛ is carried out in analogous manner. Let Σ = R × [0, 1] ∪˙ R × [1, 2]/∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 1−) ∼ (s, 1+) and (s, 0) ∼ (s, 2)

for s ≤ 0, for s ≥ 0.

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Figure 2. The pair of pants

This endows Σ with the structure of a smooth, noncompact Riemann surface with an obvious holomorphic 2-sheeted branched cover p: Σ → R × T, [s, t] → (s, t mod 1), with a single branch point on (0, 0) ∼ (0, 1) ∼ (0, 2). As before, √ holomorphic coordinates near [0, 1] are provided by the z-map. Choose generic nondegenerate perturbations of H (r) and H (s) , again denoted by H, K, such that H(1) = K(0) and such that x # y  PΛ (H # K) for any x ∈ P(H), y ∈ P(K) with x(1) = y(0). ∞ ∗ Again, after rescaling we identify PΛ 2 (H # K) ⊂ C (R/2Z, T Q) with a set of Λ Λ Λ 1-periodic loops. Given x ∈ P1 (H), y ∈ P1 (K) and z ∈ P2 (H # K) we set 
 Λ ∗ M x,y;z = u: Σ → T Q smooth u s − J(t, u) ut − XH (t, u) = 0, ∀ (s, t) ∈ Σ,  x(t), 0 ≤ t ≤ 1−, lim u(s, t) = y(t), 1+ ≤ t ≤ 2, s→−∞  lim u(s, t) = z(t), 0 ≤ t ≤ 2 . s→∞

Analogously to Proposition 3.1, for generic J, MΛx,y;z is a manifold of dimension dim MΛx,y;z = µΛ (x) + µΛ (y) − µΛ (z) − n. (10) We have also the analogous compactness result as in Proposition 3.2, and thus we can define Λ Λ mΛ : CFΛ k (H) ⊗ CFl (K) → CFk+l−n (H # K), with

Λ Λ
Λ Λ ∂Λ H#K ◦ m = m ◦ ∂H ⊗ idK ± idH ⊗ ∂K .

(11)

Here, we point out that the above branched-cover definition of the pair-of-pants domain Σ is essential for the sharp energy estimate
|∂ s u|2J ds dt ≥ 0, AH (x) + AK (y) − AH#K (z) = Σ

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

Figure 3.

89

The cylinder with slit

and it is crucial for the C 0 -estimate for solutions in the noncompact space T ∗ Q, where H and K have to satisfy (H1 ) and (H2 ). Also again, mΛ is compatible with the R-filtration of HF∗Λ . 3.2. THE RING HOMOMORPHISMS BETWEEN FREE LOOP SPACE FLOER HOMOLOGY AND BASED LOOP SPACE FLOER HOMOLOGY AND CLASSICAL HOMOLOGY

3.2.1. The homomorphism h Let us now give the construction of the homomorphism

Λ Ω h: HFΛ → HFΩ ∗ ,m ∗−n , m . We define the underlying domain as Σ = R × [0, 1]/∼ with (s, 0) ∼ (s, 1) for s ≤ 0, i.e., the interior Σ˚ is given by the open subset R × T \ [0, ∞) × {0} of the cylinder R × R/Z with the standard complex structure. Again, Σ carries the structure of a Riemann surface with boundary, the holomorphic boundary chart at (0, 0) again √ given by the z-map. We now assume H: T × T ∗ Q → R to be simultaneously nondegenerate for Ω closed orbits PΛ 1 (H) as well as for the open orbits P1 (H) based at q0 ∈ Q. Moreover, we assume that x(0)  T q∗0 Q for all x ∈ PΛ 1 (H). The corresponding Floer-trajectory solution set is defined as  Λ→Ω ∞ ˚ ∗ 0 ∗ M x;y (J, H) = u ∈ C (Σ, T Q) ∩ C (Σ, T Q)
 u s − J ut − XH (t, u) = 0, lim u(s, ·) = x, s→−∞  ∗ lim u(s, ·) = y, u(s, 0), u(s, 1) ∈ T q0 Q, ∀ s ≥ 0 . s→∞

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PROPOSITION 3.3. In view of the nondegeneracy assumption on H and q0 , it holds that (a) for generic J, MΛ→Ω is a smooth manifold of dimension µΛ (x) − µΩ (y) − n, x;y (b) it is compact in dimension 0 and AH (x) ≥ AH (y) whenever it is nonempty, and (c) it is compact in dimension 1 up to simple splitting of Floer trajectories of type Λ respectively Ω at the respective end of Σ. As before in (7) and (11) we define h on the chain level by  Ω h: CFΛ h(x) = x, yΛ→Ω y, ∗ → CF∗−n ,

(12)

y



with x, y

Λ→Ω

=

Λ Ω #alg MΛ→Ω x;y , µ (x) − µ (y) = n, 0, else,

Ω Ω Λ for x ∈ PΛ 1 (H), y ∈ P1 (H). By Proposition 3.3(c) we have ∂ ◦ h = h ◦∂ and hence, h is well-defined on homology level. We also have a sharp energy estimate Λ Λ→Ω  ∅, so that compatibility which implies that AΩ H (y) ≤ AH (x) whenever M x;y with the R-filtration follows,

h: HFΛ,≤a (H) → HFΩ,≤a ∗−n (H). ∗

(13)

PROPOSITION 3.4. The two operators h ◦mΛ

and

mΩ ◦ (h ⊗ h): CFΛ,≤a (H) ⊗ CFΛ,≤b (K) → CFΩ,≤a+b k l k+l−2n (H # K)

for generic H, K satisfying the conditions for Proposition 3.2, (11) and Proposition 3.3 are chain-homotopy equivalent. Consequently,

Λ Ω h: HFΛ → HFΩ ∗ ,m ∗−n , m is a homomorphism compatible with mΛ and mΩ . We briefly sketch the key ideas of the proof. Composing h with mΛ corresponds to a gluing result for the solutions spaces MΛx,y;z (H # K) and Λ→Ω (H # K) for x ∈ PΛ (H), y ∈ PΛ (K), z ∈ PΛ (H # K) and z ∈ PΩ (H # K) Mz;z 1 1 2 2 Λ→Ω is rescaled to width 2. This defines a 1 – 1 where the domain Σ for Mz;z correspondence of  ˙ Λ→Ω MΛx,y;z × Mz,z µΛ (z)=k+l−n

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

−R

−R

R Figure 4.

The glued domain

91

R

ΣIR

with a set MIx,y;z (R) of solutions u: ΣIR → T ∗ Q,


 u s − J(u) ut − XH (t, u) = 0,

on a domain ΣIR = R × [0, 1] ∪˙ R × [1, 2]/ ∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 0) ∼ (s, 2) and (s, 1−) ∼ (s, 1+) (s, 1−) ∼ (s, 1+)

for s ≤ −R, for − R ≤ s ≤ R, for s ≥ R,

satisfying u(s, 0), u(s, 2) ∈ T q∗0 Q for s ≥ R and  x(t), 0 ≤ t ≤ 1−, and lim u(s, t) = y(t), 1+ ≤ t ≤ 2, s→−∞

lim u(s, t) = z(t), 0 ≤ t ≤ 2,

s→∞

provided that R ≥ R(x, y, z) is large enough. This latter solution set MIx,y;z (R) is again a 0-dimensional compact, oriented manifold, and, up to simple splitting of Λ- and Ω-Floer trajectories, it is compactly cobordant to a set MIIx,y;z (R ) of solutions
 u: ΣIIR → T ∗ Q, u s − J(u) ut − XH (t, u) = 0, on a domain ΣIIR = R × [0, 1] ∪˙ R × [1, 2]/∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 1−) ∼ (s, 1+)

for s ≤ −R , for s ≥ R ,

satisfying u(s, 0), u(s, 2) ∈ T q∗0 Q u(s, 1−), u(s, 1+) ∈ T q∗0 Q

for s ≥ −R for − R ≤ s ≤ R .

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−R

−R

R Figure 5.

R

The glued domain ΣIIR

The cobordism between MI and MII is obtained by a family of domains realizing a continuous deformation of ΣIR into ΣIIR . Again, for R ≥ R (x, y, z) large enough, the solutions in MIIx,y;z (R ) are in 1 – 1 correspondence with the disjoint union  Λ 

˙ 
Λ→Ω Λ→Ω Ω Λ M x;x (H) × My;y (K) × M x ,y ;z (H # K) µ (x ) = k − n, µ (y ) = l − n by a second gluing argument. The fact, that the cobordism between MIx,y;z (R) and MIIx,y;z (R ) is only compact up to simple splitting of Floer trajectories, is responsible for h ◦mΛ and mΩ ◦ (h ⊗ h) being only chain-homotopy equivalent instead of being equal. 3.2.2. Construction of e with right-inverse i In order to carry out all constructions on the chain level, we represent the classical homology of Q by means of Morse homology with auxiliary Morse functions f ∈ C ∞ (Q, R). Also the intersection product • is defined on chain level, namely by means of the trivalent-graph construction. This means that we define a chain complex (CM∗ ( f ), ∂) by CM∗ ( f ) = Crit∗ f ⊗ Z,  #alg (M x,y ( f, g)/R)y, ∂x = i(y)=i(x)−1

where the set of nondegenerate critical points Crit∗ f is Z-graded by the Morse index i(·), and ( f, g) is a Morse – Smale pair of Morse function f and Riemannian metric g such that transversality holds, and thus 


M x,y ( f, g) = γ: R → Q γ˙ (s) + ∇g f γ(s) = 0, γ(−∞) = x, γ(+∞) = y = W u (x)  W s (y). is a manifold of dimension i(x) − i(y). We have ∂2 = 0 and def

H∗ (Q)
HM∗ ( f ) = (CM∗ ( f ), ∂) = ker ∂/ im ∂.

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93

A homotopy of Morse functions f f provides a canonical continuation isomorphism H∗ ( f )
H∗ ( f ). For the definition of the intersection product, we choose a generic triple of Morse functions f , f , f such that Crit f ∩ Crit f = ∅. For x ∈ Critk f , y ∈ Critl f and z ∈ Critm f we set 
 M x,y;z ( f, f , f , g) = γ, γ ∈ C ∞ (−∞, 0], Q , γ ∈ C ∞ ([0, ∞), Q) γ˙ + ∇g f (γ) = 0, γ˙ + ∇g f (γ ) = 0, γ˙ + ∇g f (γ ) = 0,

γ(−∞) = x, γ (−∞) = y, γ (+∞) = z, γ(0) = γ (0) = γ (0) . For a generic Riemannian metric g, M x,y;z ( f, f , f , g) is a manifold of dimension i(x) + i(y) − i(z) − n, compact in dimension 0 and compact up to simple splitting in dimension 1, such that counting elements of M x,y;z with orientation defines •: CMk ( f ) ⊗ CMl ( f ) → CMk+l−n ( f ),

(14)

which induces the intersection product of Q on homology level. Note that here, the orientability of Q is required for the existence of a coherent orientation of the solution sets M x,y;z . Let us now consider the half-cylinders Z + = [0, ∞) × T,

Z − = (−∞, 0] × T,

and define for x ∈ PΛ 1 (H) the space 
 M−x (J, H) = u ∈ C ∞ (Z − , T ∗ Q) u s − J(u) ut − XH (t, u) = 0, lim u(s, t) = x(t), s→−∞

u(0, t) ∈ OQ ⊂ T ∗ Q, ∀ t ∈ T

(15)

for generic J as above under the generic assumption on H that no x ∈ PΛ 1 (H) lies completely within the zero section OQ . Since the zero section is a Lagrangian submanifold with vanishing Maslov class, µMaslov (c) = 0 for all c ∈ ΛQ, we obtain that M−x is a smooth manifold of dimension dim M−x (J, H) = µΛ (x). Similarly, we define for any q ∈ Q and x ∈ PΛ 1 (H) the set 
 M+q;x (J, H) = u ∈ C ∞ (Z + , T ∗ Q) u s − J(u) ut − XH (t, u) = 0, lim u(s, t) = x(t), s→+∞

u(0, t) ∈ T q∗ Q, ∀ t ∈ T

(16)

which, if nonempty, represents a manifold of dimension µΛ (x) for J generically chosen for the fixed q ∈ Q, provided that no x ∈ PΛ 1 (H) lies entirely within the Lagrangian fiber T q∗ Q.

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Consider now y ∈ Crit f with W u (y), W s (y) the unstable and stable manifolds of the negative gradient flow of −∇g f on Q. We set M−x;y (H; f ) = {u ∈ M−x (H) | u(0, 0) ∈ W s (y)}, M+y;x ( f ; H) = {(q, u) | q ∈ W u (y), u ∈ M+q;x (H)}.

(17)

For generic J and g, these spaces are manifolds of dimensions dim M−x;y = µΛ (x) − i(y),

dim M+y;x = i(y) − µΛ (x).

They are compact in dimension 0 and compact, up to simple splitting of either ΛFloer trajectories on one end or Morse-trajectories on the other end, in dimension 1. Hence, we obtain on chain level well-defined group homomorphisms 
 e: CFΛ (H) → CM ( f ), e(x) = #alg M−x;y (H, f ) y, k k i(y)=k

i: CMk ( f ) →

CFkΛ (H),

i(y) =




 #alg M+y;x ( f, H) x.

(18)

µΛ (x)=k

It can now be shown that e ◦ i is chain-homotopy equivalent to idCM∗ ( f ) . The main argument is that the intermediate 0-dimensional solution space ' (R) = u: [−R, R] × T → T ∗ Q u − J(u)
u − X (t, u) = 0, M y,y

s

t

H T q∗ Q, q

u({−R} × T) ⊂ ∈ W u (y),

u({+R} × T) ⊂ OQ , u(R, 0) ∈ W s (y ) is cobordant to W u (y)∩W s (y ) via a homotopy H we have for (−J)-holomorphic annuli

0. Namely, by Stokes’ theorem

u: [−R, R] × T → T ∗ Q, u s − J(u)ut = 0, u({−R} × T) ⊂ T q∗ Q, u({+R} × T) ⊂ OQ by compatibility of J with ω that  R 1
2 2 1 E(u) = 2 (|u s | J + |ut | J ) ds dt = − u∗ ω −R 0 [−R,R]×T   u(−R, ·)∗ θLiouv − u(+R, ·)∗ θLiouv = 0, = T

T

(19)

because θLiouv|OQ = 0 = θLiouv|T q∗ Q , and hence u = const. This proves that e ◦ i idCM∗ ( f ) . Similarly, one can establish a suitable cobordism between the solution space 'I (R) = u: Σ− → T ∗ Q u − J(u)
u − X (t, u) = 0, M x,x ;y

R

s

t

H

u(−∞, ·)|[0,1−] = x, u(−∞, ·)|[1+,2] = x ,


u {R} × R/2Z ⊂ OQ , u(R, 0) ∈ W s (y; f ) ,

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where Σ−R = {(s, t) ∈ Σ | s ≤ R}, R > 0, for the pair-of-pants domain Σ for mΛ from Section 3.1.2, and the solution space 'II (R) = {(u, u , γ, γ ) | u ∈ M− (J, H), u ∈ M− (J, H ), M x x,x ;y x γ, γ : [−R, 0] → Q, γ˙ = −∇g f (γ), γ˙ = −∇g f (γ ), γ(−R) = u(0, 0), γ (−R) = u (0, 0), γ(0) = γ (0) ∈ W s (y; f )} for the Morse functions f , f , f from (14). This is the main argument for proving that on chain level, the compositions e ◦mΛ

Λ Λ • ◦ (e ⊗ e): CFΛ k (H) ⊗ CF l (K) → CFk+l−n ( f )

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are chain homotopy equivalent. From these considerations we obtain PROPOSITION 3.5. For the linear operators e: HFΛ ∗ → H∗ (Q) and i: H∗ (Q) → Λ : HFΛ ⊗ HFΛ → HFΛ and •: H (Q) ⊗ H (Q) HFΛ , and the bilinear operators m ∗ ∗ ∗ ∗ ∗ ∗−n → H∗−n (Q) we have (a) e ◦ i = idH∗ (Q) ,
 (b) e mΛ (α, β) = e(α) • e(β), for all α, β ∈ HFΛ ∗. This completes the exposition of the arguments for statements (a) and (b) in Theorem 2.7 and the upper row in the commutative diagram. In the following section, we give details on the remaining vertical isomorphisms in that diagram which gives the main statement of the theorem, namely that Floer homology in the cotangent bundle encodes the well-known loop space homology. 4. Morse-homology on the loop spaces ΛQ and ΩQ, and the isomorphism In this section, we give a short account of the construction of the explicit isomorphisms ΦΛ and ΦΩ from Theorem 2.7. These isomorphisms have been established in Abbondandolo and Schwarz (2004), by representing the classical homology of the loop space in terms of Morse homology for the Lagrangian action functional. The central idea of the isomorphism is the Legendre transform as a duality between the Lagrangian and the Hamiltonian picture. Different approaches to the same isomorphism of homology groups had been established first by Viterbo (1996) and by Salamon and Weber (2003). Let us consider the connected, compact, oriented smooth manifold Q with an auxiliary Riemannian metric g and an admissible Lagrangian function L: T × T Q → R such that the following assumptions hold: For any coordinate chart
 U, (q1 , . . . , qn ) on Q and the associated local trivialization of T Q|U there exist constants l0 > 0, l1 ≥ 0 such that we have in the coordinates (q1 , . . . , qn , v1 , . . . , vn )

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(L1 ) ∂2vv L(t, q, v) ≥ l0 as a quadratic form, (L2 ) |∂2qq L(t, q, v)| ≤ l1 (1 + |v|2 ), |∂2qv L(t, q, v)| ≤ l1 (1 + |v|), |∂2vv L(t, q, v)| ≤ l1 for all t ∈ T, q ∈ U, v ∈ T q Q. These assumptions are formally similar to (H1 ) and (H2 ). They imply that the Lagrangian L grows fiberwise at least quadratically and at most quadratically. By the strict convexity condition (L1 ) we have the Legendre transform


LL : T × T Q − → T × T ∗ Q, 
v TQ , (t, q, v) → (t, q, dL(t, q, v)|T (q,v)

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v T Q = ker Dτ = T (T Q) = T Q can be canonically identified, and where T (q,v) v q q τ: T Q → Q is the bundle projection. Using the inverse transform ∗ L−1 L : T × T Q → T × T Q, (t, q, p) → (t, q, v(t, q, p)),

we have the Hamiltonian function H(t, q, p) associated to L(t, q, v) by
 H(t, q, p) = max p, v − L(t, q, v) v∈T q Q

= p, v(t, q, p) − L(t, q, v(t, q, p)).

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The Legendre transform LL gives a diffeomorphism by which we can also pull back the symplectic structure ωLiouv from T ∗ Q to T Q, and the Hamiltonian vector field XH is mapped to the vector field YL on T Q whose integral curves are characterized by the second order ODE in the q-variables 

∇t ∇v L(t, q(t), q(t) ˙ = ∇q L t, q(t), q(t) ˙ ,

(23)

where ∇t denotes the covariant derivation along q, and ∇v , ∇q denote the vertical and the horizontal part of the gradient of L with respect to the induced Riemannian metric from g and its Levi-Civita connection. (23) is also the Euler-Lagrange equation for the Lagrangian action functional 

1

EL (q) =


 L t, q(t), q(t) ˙ dt

0

viewed on ΛQ and, respectively, ΩQ. Its critical sets are the solution spaces PΛ (L) = {q ∈ ΛQ | q solves (23)} PΩ (L) = {q ∈ ΩQ | q solves (23)}.

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We use the same notation as in the Hamiltonian picture since by Legendre transform we have the canonical one-to-one correspondence 1:1

π: PΛ resp. Ω (H) −−→ PΛ resp. Ω (L), x = (q, p) → q. The nondegeneracy conditions for the Legendre picture read: (L0)Λ For every solution q ∈ PΛ (L) there exists no 1-periodic Jacobi vector field along q, (L0)Ω for every solution q ∈ PΩ (L) there exists no Jacobi vector field along q vanishing for t = 0 and t = 1. These conditions are equivalent to the nondegeneracy conditions for x = π−1 (q) ∈ PΛ resp. Ω (H) as solutions of the Legendre dual Hamiltonian problem, i.e., (H0)Λ the differential of the time-one integral map of XH at x(0) does not have the eigenvalue 1, (H0)Ω the differential of the time-one integral map of XH , T x(0) T ∗ Q → T x(1) T ∗ Q, maps the vertical subspace of x(0) onto a subspace having intersection {0} with the vertical subspace at x(1). The crucial observation for the isomorphism between Morse homology for EL in the Lagrangian picture and Floer homology for AH in the Hamiltonian picture is the LEMMA 4.1 (Main lemma). If x = (q, p): [0, 1] → T ∗ Q is continuous, with q of class W 1,2 , then AH (x) ≤ EL (q), v T Q , that is if and only the equality holding if and only if p(t) = dL(t, q(t), q(t))| ˙ T q(t) ˙ ˙ for every t ∈ [0, 1]. In particular, the Hamilif (t, q(t), p(t)) = LL (t, q(t), q(t)) tonian and the Lagrangian action functionals coincide on the solutions of the two systems.
 ∗ Q that Proof. From the definitions we have for x(t) = q(t), p(t) ∈ T q(t)

AH (x) =

 1


 p(t), q(t) ˙ − H t, q(t), p(t) dt



0 1

=  ≤ 0

(p(t), q(t) ˙ − max[p(t), v − L(t, q(t), v)]) dt v

0 1


 L t, q(t)q(t) ˙ dt = EL (q),

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by choosing pointwise v = q(t), ˙ with equality exactly if (t, q, p) = LL (t, q, q). ˙




On the canonically identified solution sets PΛ resp. Ω (H) = Crit AH and = Crit EL we have not only the same critical values for AH and EL by the previous lemma, we also can identify the Conley – Zehnder index µΛ and the Maslov index µΩ on the Hamiltonian side with the Morse indices iΛ and iΩ of EL on the Lagrangian side. PΛ resp. Ω (L)

THEOREM 4.2 (Duistermaat, 1976; Robbin and Salamon, 1993; Weber, 2002).
 Assuming that Q is orientable, let q ∈ PΛ (L), and let x ∈ PΛ (H), t, x(t) = LL (t, q(t), q(t)), ˙ be the corresponding 1-periodic solution of the Hamiltonian ∗ system on T Q. Then iΛ (q) = µΛ (x).

 Let q ∈ PΩ (L) and let x ∈ PΩ (H), t, x(t) = LL t, q(t), q(t) ˙ , be the corresponding solution of the Hamiltonian system on T ∗ Q. Then iΩ (q) = µΩ (x). The crucial property allowing the construction of Morse homology for EL is the Palais – Smale property which had been proven by V. Benci, THEOREM 4.3 (Benci, 1986). Consider standard W 1,2 -metrics on ΛQ and ΩQ, and let L satisfy the conditions (L1 ) and (L2 ). Then EL has the Palais – Smale property, i.e, for any sequence (qn ) ⊂ ΛQ respectively ΩQ with (EL (qn )) bounded and gradW 1,2 EL (qn ) → 0 there exists a convergent subsequence of (qn ). As a consequence, see for instance Abbondandolo and Majer (2005) in this volume, we have well-defined Morse homology for EL . That is, we have the DEFINITION 4.4. Given a, b ∈ Crit EL ⊂ ΛQ, we have 


 W u (a) = γ ∈ C ∞ (−∞, 0], ΛQ γ(−∞) = a, γ˙ = −∇EL (γ) , 

W s (b) = γ ∈ C ∞ ([0, ∞), ΛQ) γ(+∞) = b, γ˙ = −∇E (γ) , L

where ΛQ is endowed with a W 1,2 -structure and the gradient ∇EL is taken with respect to a Hilbert-metric on ΛQ as W 1,2 -Hilbert manifold. We have W u (a) ∩ W s (b) = MΛ a,b (EL ) = {γ: R → Λ | γ˙ + ∇EL (γ) = 0, γ(−∞) = a, γ(+∞) = b}, and completely analogously for the corresponding negative gradient flow on ΩQ.

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Again, one can show that for a generic perturbation of the Hilbert metric on ΛQ respectively ΩQ, we have transversality W u (a)  W s (b), such that this trajectory space is a smooth, finite-dimensional manifold of dimension Λ resp. Ω

dim Ma,b

(EL ) = iΛ resp. Ω (a) − iΛ resp. Ω (b).

Moreover, due to the Palais – Smale property of EL , the trajectory spaces Ma,b have the required compactness properties up to splitting of trajectories. We then define CM∗ (EΛ resp. Ω ) = Crit EL ⊗ Z, which are Z-graded by iΛ respectively iΩ and endowed with the R-filtration CMa∗ (EΛ resp. Ω ) = {x ∈ Crit EL | EL (x) ≤ a} ⊗ Z. Using the concept of Morse homology for EL , we define chain boundary operators Λ resp. Ω ∂ on CM∗ (EL ) by counting Ma,b (El ) as before and we obtain the Morse homology HM∗ (EΛ resp. Ω ) = H∗ (CM∗ (EΛ resp. Ω ), ∂) and likewise for the filtered complex (CMa∗ , ∂). Recalling from Section 2.1 that the inclusions of the Hilbert manifolds W 1,2 (T, Q) and {c ∈ W 1,2 ([0, 1], Q) | c(0) = c(1) = q0 } into Λ0 Q = C 0 (T, Q)
 and Ω0 Q = C 0 ([0, 1], {0, 1}), (Q, q0 ) are homotopy equivalences, we obtain THEOREM 4.5. For all a ∈ R there exist canonical isomorphisms Λ resp. Ω

HMa∗ (EL

Λ resp. Ω

)
H∗ ({EL

Λ resp. Ω HM∗ (EL )

≤ a}),


H∗ (Λ Q resp. Ω0 Q), 0

compatible with the induced homomorphisms from {E ≤ a} ⊂ {E ≤ b} for a ≤ b, and lim HMa∗ (EL ) = HM∗ (EL ). −→ We now show that this Morse homological representation by the Lagrangian action functional is directly isomorphic to the Floer homological representation based on the Legendre dual Hamiltonian action functional. THEOREM 4.6. Assume that L is nondegenerate and satisfies (L1 ), (L2 ) above and that H is the Legendre transform of L. Let J be a sufficiently small generic perturbation of JLC for some Riemannian metric g0 on Q, and let g be a generic Riemannian structure on the Hilbert manifolds Λ1 Q and Ω1 close to ·, ·1,2 from g0 . Then, there exists a chain complex isomorphism for both loop space cases Λ and Ω, 


→ CF∗ (H), ∂(H, J) Θ: CM∗ (EL ), ∂(L, g) −

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of the form Θ(q) =



d+ (q, x) · x

∀ q ∈ P(L)

x∈P(H),µ(x)=i(q)

with d+ (q, x) = 0 whenever EL (q) ≤ AH (x) except for d+ (q, x) = ±1 if π(x) = q. In particular, Θ is also an isomorphism between the restricted chain complexes CMa∗ (EL ) and CFa∗ (H) for a ∈ R. We remark that the conditions (H1 ) and (H2 ) are automatically satisfied for H = LL (L) if L satisfies (L1 ) and (L2 ). In the following we sketch the proof of Theorem 4.6 which is given in full terms in Abbondandolo and Schwarz (2004). For simplicity, we drop the notational distinction between ΛQ and ΩQ which are considered completely analogously. Proof. Recall first that the generators of CM∗ (EL ) and CF∗ (H) allow a canonical identification by P(L) = Crit EL  q ↔ x ∈ Crit AH = P(H), via π(x) = q where π: T ∗ Q → Q is the cotangent bundle projection, and where EL (q) = AH (x) for those critical points, in view of the Legendre duality. Choose
 a numbering (qn )n∈N of Crit EL such that EL (qn ) is monotone nondecreasing, and let (xn ) be the corresponding sequence in Crit AH . The crucial issue is the observation that the operator Θ is represented in terms of this infinite basis by a
 matrix D = d+ (qi , x j ) i, j∈N which has the properties 1. D is lower-triangular, i.e., d+ (qi , x j ) = 0 for i < j, 2. dii+ = ±1 for all i ∈ N, 3. DE = A D, where E and A are the matrices representing the chain boundary operators ∂E : CM∗ → CM∗−1 and ∂A : CF∗ → CF∗−1 . It is an immediate consequence from
→ (CF∗ , ∂A ) is 1 and 2 that D is invertible and with 3 we obtain that Θ: (CM∗ , ∂E ) − a chain complex isomorphism. It remains to give the construction of d+ (q, x) for q ∈ P(L), x ∈ P(H) such that 1, 2 and 3 are satisfied. Let q ∈ Crit EL and x ∈ Crit AH and define  ME→A = (γ, u) ∈ W u (q) × C ∞ ([0, ∞) × T, T ∗ Q) q;x
 u s − J ut − XH (t, u) = 0, u(+∞) = x,


π u(0, t) = γ(0)(t) ∀ t ∈ T . (24) Here, we use the loop γ(0) in the finite-dimensional   unstable manifold as a 1∗ parameter family of Lagrangian subspaces T γ(0)(t) Q providing a t-dependent t∈T family of boundary conditions for the pseudoholomorphic half-cylinder u where

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the boundary condition identifies the loop parameter for u(0) with the family parameter. This is again an admissible nonlinear Fredholm problem, for which we can prove PROPOSITION 4.7. is a (a) For generic J and g as in Theorem 4.6, the mixed solution space ME→A q;x smooth manifold of dimension i(q) − µ(x). is compact in dimension 0 and compact, up to simply splitting of (b) ME→A q;x either Morse trajectories on the negative end or Floer trajectories on the positive end, in dimension 1. (c) We have ME→A  ∅ only if EL (q) ≥ AH (x) with EL (q) = AH (x) if and only q;x 

= (q, x) , i.e. γ(0) = q and u(0) = x in (24) above. if π(x) = q and ME→A q;x Defining now d+ (q, x) = #alg ME→A by counting with signs, which are q;x determined by a concept of coherent orientations described completely in (Abbondandolo and Schwarz, 2004), we obtain the properties 1 – 3 as claimed. Concerning this proposition, the essential new argument is based on Main Lemma 4.1. Namely, for any (γ, u) ∈ ME→A we have EL (γ(0)) ≥ AH (u(0)) q;x and thus EL (q) ≥ AH (x) with equality if and only if γ and u are constant with respect to the flow parameter. This implies an a priori energy bound required for the compactness statements in (b). Moreover, we see that d+ (qi , x j ) = 0 for i < j with respect to the above ordering of the generators. Note that the transversality property of (a) in case of the constant solution (q, x) ∈ ME→A cannot be achieved by perturbing J or g. In fact, we can prove q;x that for π(x) = q we have automatic transversality which follows from the second variation estimate d2 AH (x)[ζ, ζ] ≤ d2 EL (q)[Dπ(x)ζ, Dπ(x)ζ],

for all ζ ∈ T x Λ(T ∗ Q).

This estimate is again a consequence of Main Lemma 4.1. For details, see Abbondandolo and Schwarz (2004). Thus, the core argument is based on Legendre duality in an essential way. Another important remark is that there is no analogous construction of the inverse chain homomorphism Θ−1 : CF∗ → CM∗ via a correspondingly reversed mixed solution space. If we consider the problem  
= (u, γ) ∈ C ∞ (−∞, 0] × T, T ∗ Q × W s (q) u s − J(ut − XH (t, u)) = 0, MA→E x;q


u(−∞) = x, π u(0) = γ(0) ,

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there is no possibility to obtain an analogous statement as in Proposition 4.7. Both, the Fredholm property and the compactness property will fail in general, because dim W s (q) = ∞ and EL (γ(0) ≥ AH (u(0)) which allows no a priori energy bound.
5. Products in Morse-homology In this last section, we sketch the Morse-homological definitions of the product and intersection operations o, ∩ and # from Section 2.1. These are needed in order to prove that the explicit isomorphism Θ intertwines the product structures on both sides as claimed in Theorem 2.7. The finite-dimensional model of the homomorphism ∩ in Morse homology is given by the following construction. Let M m , N n be smooth, finite-dimensional manifolds, for simplicity closed. Let i: M → N be a smooth cooriented embedding and f M : M → R, f N : N → R be Morse functions such that Crit f N ∩ i(M) = ∅.

(25)

Then we obtain induced homomorphisms of Morse homology, i∗ : HM∗ (M, f M ) → HM∗ (N, f N ), i! : HM∗ (N, f N ) → HM∗−(n−m) (M, f M ), where i! equals ∩ from Section 2.1. The construction goes as follows on chain level. By assumption (25), which is generically fulfilled, we can find generic Riemannian metrics g M , gN on M and N such that we have the transverse intersections  i(M)  W s (x; f N , gN ) = γ: [0, ∞) → N γ˙ + ∇gN f N (γ) = 0,

γ(0) ∈ i(M), γ(∞) = x ,  i(M)  W u (x; f N , gN ) = γ: (−∞, 0] → N γ˙ + ∇gN f N (γ) = 0,

γ(0) ∈ i(M), γ(−∞) = x , (26) for all x ∈ Crit f N , and ∗ Mix,y = W u (x; f M , g M ) i W s (y; f N , gN )
  = (γ M , γ N ) ∈ C ∞ (−∞, 0], M × C ∞ ([0, ∞), N)

! Miy,x

γ˙ K + ∇gK f K (γ K ) = 0, K ∈ {M, N},


γ M (−∞) = x, γ N (+∞) = y, i γ M (0) = γ N (0) , = W u (y; f N , gN ) i W s (x; f M , g M )  = (γ N , γ M ) ∈ C ∞ ((−∞, 0], N) × C ∞ ([0, ∞), M) γ˙ K + ∇gK f K (γ K ) = 0, K ∈ {M, N}, γ N (−∞) = y, γ M (+∞) = x, i(γ M (0)) = γ N (0)



NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY


= W u (y; f N , gN ) N i(M)  M W s (x; f M , g M ),

103 (27)

with ∗ = i f M (x) − i f N (y), dim Mix,y

! dim Miy,x = i f N (y) − i f M (x) − (n − m),

for x ∈ Crit f M and y ∈ Crit f N . Here,  M , N indicate the ambient manifold of the transversely intersecting submanifolds. Then, counting the solution of (27) in dimension 0 defines i∗ and i! on chain-level. Let us use this concept now in the infinite-dimensional Morse-homological framework of the codimension-n embedding i: Ωq0 Q → ΛQ of the Hilbert manifolds, where the Morse function is given by the Lagrangian action functional for some nondegenerate Lagrangian L satisfying (L1 ) and (L2 ). The transversality condition (25) is now given by the assumption that for all c ∈ PΛ (L) we have c(0)  q0 , i.e., PΛ (L) ∩ i(Ωq0 Q) = ∅,

(28)

which is fulfilled generically for (L, q0 ). Choosing generic Riemannian metrics gΛ and gΩ on ΛQ and ΩQ we construct Λ i∗ : HM∗ (EΩ L ) → HM∗ (EL )

on chain level by



i∗ (a) =


#alg W u (a; EΩ , gΩ )  W s (b; EΛ , gΛ ) · b

b∈PΛ (L),i(b)=i(a)

for all a ∈ Crit EΩ L , where the intersection means counting γ: R → ΛQ such that γ˙ (s) = −∇gΩ EΩ (γ) and γ(s)(0) = q0 for all s ≤ 0 and γ˙ (s) = −∇gΛ EΛ (γ) for all s ≥ 0. Similarly, we define Ω i! : HM∗ (EΛ L ) → HM∗−n (EL )

on chain level by i! (a) =




  s Ω Ω  #alg W u (a; EΛ , gΛ )  ev−1 0 (q0 )  W (b; E , g ) · b

b∈PΩ (L),i(b)=i(a)−n

for a ∈ PΛ , where the double intersection means counting γ− : (−∞, 0] → ΛQ and γ+ : [0, ∞) → ΩQ such that γ˙ ± = −∇gΛ resp. Ω EΛ resp. Ω (γ± ), γ− (−∞) = a, γ+ (∞) = 
b, ev0 γ− (0) = q0 and γ− (0) = γ+ (0). One can show that, assuming (28), all intersections are transverse for generic gΛ , gΩ .

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Next we consider the codimension-n embedding,
 #: ΩQ × ΩQ → Ω2 Q ⊂ C 0 [0, 2], Q ,  c(t), 0 ≤ t ≤ 1, #(c, d)(t) = d(t), 1 ≤ t ≤ 2, which for convenience we view into the based loop space of period 2. The generic assumption on (L, q0 ) for transversality is now that c(1)  q0

for all c ∈ PΩ 2 (L).

(29)

Then, for generic Riemannian metrics g1 , g2 on ΩQ on g3 on Ω2 Q we have 
# W u (a; EΩ , g1 ) × W u (b; EΩ , g2 )  W s (c; EΩ2 , g3 ) Ω for all a, b ∈ PΩ 1 (L), c ∈ P2 (L). Counting in dimension 0 with signs thus defines on chain level an operator which on homology gives      Ω Ω 2 #∗ : HMk EΩ L ⊗ HMl EL → HMk+l EL .

Let us now turn to the codimension-n submanifold #

X = {(c, d) ∈ ΛQ × ΛQ | c(0) = d(0)} → Λ2 Q = W 1,2 (R/2Z, Q). In order to construct

      Λ Λ i! : HMk EΛ L1 ⊗ HMl EL2 → HMk+l−n X; EL1 #L2

and

   Λ  → HM EL12#L2 #∗ : HMk+l−n X; EΛ k+l−n L1 #L2

we assume a(0)  b(0) cX

for all (a, b) ∈ PΛ (L1 ) × PΛ (L2 ), for all c ∈ PΛ 2 (L1 # L2 ),

(30)

where L1 (1, ·) = L2 (0, ·), L = L1 # L2 : R/2Z × T Q → R, with L|[0,1] = L1 and L|[1,2] = L2 . By assumption (30) we can find generic Riemannian metrics g1 , g2 on ΛQ and g3 on Λ2 Q such that we have transverse intersections      u Λ × W b; E  X, , g , g Y = W u a; EΛ 1 2 L1 L2   2 Z(a, b; c) = #(Y)  W s c; EΛ L1 #L2 , g3 for all (a, b) ∈ PΛ (L1 ) × PΛ (L2 ) and c ∈ PΛ2 (L1 #L2 ). We define      Λ  Λ → CM ⊗ CM E EL12#L2 , o: CMk EΛ l k+l−n L1 L2 aob= #alg Z(a, b; c) · c, i(c)=i(a)+i(b)−n

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NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

γ1

γ3

105

u

γ2 0 Figure 6.

R The domain for MIa,b;x (R)

which on homology gives o = # ◦ i! , the Morse-homological definition of the loop product. 5.1. RING ISOMORPHISM BETWEEN MORSE HOMOLOGY AND FLOER HOMOLOGY

Finally, we sketch very briefly the proof of the ring homomorphism property of   
     Ω Ω , # A , m Θ∗ : HM∗ EΩ − → HF ∗ ∗ L H in the based loop space case. Full details, also for the free loop space case, will be contained in Abbondandolo and Schwarz (2005). We show that on chain level, the compositions of Θ and # and mΩ and Θ, respectively, are chain homotopy equivalent, Θ◦#

Ω2 Ω mΩ ◦ (Θ ⊗ Θ): CMΩ ∗ (EL ) ⊗ CM∗ (EL ) → CF∗ (AH ).

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Consider a generic triple of Riemannian metrics g1 , g2 on ΩQ and g3 on Ω2 Q as before, and J generic on T ∗ Q for AH on Λ2 (T ∗ Q). Then, by gluing, the composition (Θ ◦ #∗ )(a, b) is given by counting  MIa,b;x (R) = (γ1 , γ2 , γ3 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ3 : [0, R] → Ω2 Q, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, 3, γ1 (−∞) = a, γ2 (−∞) = b, γ3 (0) = γ1 (0) # γ2 (0),
 u: [R, ∞) × [0, 2] → T ∗ Q, u s − J(u) ut − XH (t, u) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, R ≤ s < ∞,

u(+∞) = x, u(R, t) ∈ T γ∗3 (R)(t) Q, 0 ≤ t ≤ 2 for µΩ (x) = i(a) + i(b) and for R " 1 sufficiently large depending on a, b.

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γ1 u

γ2 −r Figure 7.

0 The domain for MIIa,b;x (r)

For the other composition mΩ ◦ (Θ ⊗ Θ)(a, b) we have to count for r " 1  II Ma,b;x (r) = (γ1 , γ2 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, γ1 (−∞) = a, γ2 (−∞) = b, u: Σr → T ∗ Q, u s − J(u)(ut − XH (t, u)) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, for − r ≤ s < ∞, u(s, 1±) ∈ T q∗0 Q, for r ≤ s ≤ 0,  ∗ ( T Q, 0 ≤ t ≤ 1−, u(+∞) = x, u(−r, t) ∈ γ∗1 (0)(t) T γ2 (0)(t−1) Q, 1+ ≤ t ≤ 2, Ω where x ∈ PΩ 2 (H) with µ (x) = i(a) + i(b), and Σr is the domain $ Σr = [−r, ∞) × [0, 1] ∪˙ [−r, ∞) × [1, 2] ∼,

with (s, 1−) ∼ (s, 1+) for all s ≥ 0. One can now show that both solution spaces MIa,b;x (R) and MIIa,b;x (r) are compactly, up to simple splitting, cobordant to  0 Ma,b;x = (γ1 , γ2 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, γ1 (−∞) = a, γ2 (−∞) = b, u: [0, ∞) × [0, 2] → T ∗ Q, u s − J(u)(ut − XH (t, u)) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, for 0 ≤ s < ∞,  ∗ ( T Q, 0 ≤ t ≤ 1−, u(+∞) = x, u(0, t) ∈ γ∗1 (0)(t) . T γ2 (0)(t−1) Q, 1+ ≤ t ≤ 2, This shows that
 Θ ◦ # − mΩ ◦ (Θ ⊗ Θ) = ∂Ω (J, H) ◦ S ± S ◦ id ⊗ ∂Ω (EΩ , g2 ) ± ∂Ω (EΩ , g1 ) ⊗ id for some chain homotopy operator      Ω Ω 2 S : CM∗ EΩ L ⊗ CM∗ EL → CF∗+1 AH ,

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γ1 γ2

Figure 8.

u

The intermediate domain for M0a,b;x

thus proving (32). References Abbondandolo, A. and Majer, P. (2005) Lectures on the Morse complex for infinite-dimensional manifolds, in this volume. Abbondandolo, A. and Schwarz, M. (2004) On the Floer homology of cotangent bundles, Comm. Pure Appl. Math., to appear; arXiv:math.SG/0408280. Abbondandolo, A. and Schwarz, M. (2005) Floer homology of cotangent bundles and the loop product, in preparation. Benci, V. (1986) Periodic solutions of Lagrangian systems on a compact manifold, J. Differential Equations 63, 135 – 161. Chas, M. and Sullivan, D. (1999) String topology, arXiv:math.GT/9911159. Cohen, R. (2005) Morse theory, graphs, and string topology, in this volume. Cohen, R. and Jones, J. D. S. (2002) A homotopy theoretic realization of string topology, Math. Ann. 324, 773 – 798. Cohen, R. L. and Voronov, A. A. (2005) Notes on string topology, arXiv:math.GT/0503625. Duistermaat, J. J. (1976) On the Morse index in variational calculus, Adv. in Math. 21, 173 – 195. Floer, A. (1988)a A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41, 393 – 407. Floer, A. (1988)b The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775 – 813. Floer, A. (1989) Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 575 – 611. Floer, A. and Hofer, H. (1993) Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212, 13 – 38. F´elix, Y., Halperin, S., and Thomas, J.-C. (2001) Rational Homotopy Theory, Vol. 205 of Grad. Texts in Math., New York, Springer. Gromov, M. (1985) Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82, 307 – 347. Hofer, H. and Salamon, D. A. (1995) Floer homology and Novikov rings, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, pp. 483–524, Basel, Birkh¨auser. Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and quantum cohomology, In C. B. Thomas (ed.), Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., Cambridge 1994, pp. 171–200, Cambridge, Cambridge Univ. Press. Ramirez, A. (2005), Ph.D. thesis, Stanford University, in preparation.

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Robbin, J. and Salamon, D. (1993) The Maslov index for paths, Topology 32, 827 – 844. Salamon, D. and Weber, J. (2003) Floer homology and the heat flow, arXiv:math.SG/0304383. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkh¨auser. Schwarz, M. (1995) Cohomology operations from S 1 -cobordisms in Floer homology, Ph.D. thesis, ETH Zurich. Viterbo, C. (1996) Functors and computations in Floer homology with applications. II, preprint. Weber, J. (2002) Perturbed closed geodesics are periodic orbits: index and transversality, Math. Z. 241, 45 – 82.

HOMOTOPICAL DYNAMICS IN SYMPLECTIC TOPOLOGY JEAN-FRANC¸OIS BARRAUD Universit´e de Lille 1 OCTAV CORNEA Universit´e de Montr´eal

Abstract. This is mainly a survey of recent work on algebraic ways to “measure” moduli spaces of connecting trajectories in Morse and Floer theories as well as related applications to symplectic topology. The paper also contains some new results. In particular, we show that the methods of Barraud and Cornea (2003) continue to work in general symplectic manifolds (without any connectivity conditions) but under the bubbling threshold. Key words: Serre spectral sequence, Morse theory, Lagrangian submanifolds, Hopf invariants, Hofer distance 2000 Mathematical Subject Classification: Primary 53D40, 53D12; Secondary 37D15

1. Introduction The main purpose of this paper is to survey a number of Morse-theoretic results which show how to estimate algebraically the high-dimensional moduli spaces of Morse flow lines and to describe some of their recent applications to symplectic topology. We also deduce some new applications. The paper starts with a brief discussion of the various proofs showing that the differential in the Morse complex is indeed a differential. With this occasion we introduce the main concepts in Morse theory and, in particular, the notion of connecting manifold (or, equivalently, the moduli space of flow lines connecting two critical points) which is the main object of interest in our further constructions. Moreover, an extension of one of these proofs leads naturally to an important result of John Franks (1979) which describes the framed cobordism class of connecting manifolds between consecutive critical points as a certain relative attaching map. After describing Franks’ result, we proceed to a stronger result initially proved in Cornea (2002a) which computes a framed bordism class naturally associated to the same connecting manifolds in terms of certain Hopf invariants. While these results only apply to consecutive critical points we then describe a recent method to estimate general connecting manifolds by means of the Serre spectral sequence of the path-loop fibration having as base the ambient 109 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 109–148. © 2006 Springer. Printed in the Netherlands.

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manifold (Barraud and Cornea, 2003). Some interesting topological consequences of these results are briefly mentioned as well as some other methods used in the study of these problems. The third section discusses a number of symplectic applications. We start with some results which first appeared in Cornea (2002b). These use the nonvanishing of certain Hopf invariants to deduce the existence of bounded orbits of Hamiltonian flows (obviously, inside noncompact manifolds). This is a very “soft” type of result even if difficult to prove. We then continue in Section 3.2 by describing how to use the Serre spectral sequence result to detect pseudo-holomorphic strips as well as some consequences of the existence of the strips. Most of the results of this part have first appeared in Barraud and Cornea (2003) but there are some that are new: we discuss explicitly the detection of pseudoholomorphic strips passing through some submanifold and we present a way to construct in a coherent fashion our theory for Lagrangians in general symplectic manifolds as long as we remain under a bubbling threshold. Notice that even the analogue of the classical Floer theory (which is a very particular case of our construction) has not been explicited in the literature in the Lagrangian case even if all the necessary ideas are present in some form — see Schwarz (1998) for the Hamiltonian case. The paper contains a number of open problems and ends with a conjecture which is supported by the results in Section 3.2 as well as by recent joint results of the second author with Franc¸ois Lalonde. 2. Elements of Morse theory Assume that M is a compact, smooth manifold without boundary of dimension n. Let f : M → R be a smooth Morse function and let γ: M × R → M be a negative gradient Morse – Smale flow associated of f . In particular, f is strictly decreasing along any nonconstant flow line of γ and the stable manifolds

 W s (P) = x ∈ M : lim γt (x) = P t→∞

and the unstable manifolds

 W u (Q) = x ∈ M : lim γt (x) = Q t→−∞

of any pair of critical points P and Q of f intersect transversally. One of the most useful and simple tools that can be defined in this context is the Morse complex C(γ) = (Z/2Crit( f ), d). Here Z/2S  is the Z/2-vector space generated by the set S , the vector space Z/2Crit( f ) has a natural grading given by |P| = ind f (P), ∀P ∈ Crit( f ) and d is

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the differential of the complex which is defined by  dx = ayx y |y|=|x|−1


  so that the coefficients ayx = # W u (x) ∩ W s (y) /R . This definition makes sense because the set W u (P) ∩ W s (Q) which consists of all the points situated on some flow line joining P to Q is invariant by the R-action given by the flow. Moreover, W u (P) and W s (P) are homeomorphic to open disks which implies that the set
 MQP = W u (P) ∩ W s (Q) /R has the structure of a smooth (in general, noncompact) manifold of dimension |P|−|Q|−1. We call this space the moduli space of flow lines joining P to Q. It is not difficult to understand the reasons for the noncompactness of MQP when M is compact as in our setting: this is simply due to the fact that a family of flow lines joining P to Q may approach a third, intermediate, critical point R. For this to happen it is necessary (and sufficient — see Smale, 1967, or Franks, 1979) to have some flow line which joins P to R and some other joining R to Q. This implies that when |P| = |Q| + 1 the set MQP is compact and thus the sum above is finite. For further use let’s define also the unstable sphere of a critical point P as S au (P) = W u (P) ∩ f −1 (a) as well as the stable sphere S as (P) = W s (P) ∩ f −1 (a) where a is a regular value of f . It should be noted that this names are slightly abusive as these two sets are spheres, in general, only if a is sufficiently close to f (P). In that case S au (P) is homeomorphic to a sphere of dimension |P| − 1 and S as (P) is homeomorphic to a sphere of dimension n−|P|−1. With this notations
 the moduli space MQP is homeomorphic to S au (P) ∩ S as (P) for any a ∈ f (Q), f (P) which is a regular value of f . The main properties of the object defined above are that: d2 = 0 and


 H∗ C(γ) ≈ H∗ (M; Z/2).

We will sometimes denote this complex by C( f ) and will call it the (classical) Morse complex of f . The flow γ may be in fact even a pseudo- (negative) gradient flow of f . There also exists a version of this complex over Z in which the counting of the elements in Myx takes into account orientations. There are essentially four methods to prove these properties: ) y (i) Deducing the equation y ayx az = 0 (which is equivalent to d2 = 0) from the properties of the moduli spaces Mzx with |x| − |z| = 2. (ii) Comparing ayx with a certain relative attaching map. (iii) Expressing ayx in terms of a connection map in Conley’s index theory. (iv) A method based on a deformation of the de Rham complex (clearly, in this case the coefficients are required to be in R).

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For the rest of this paper the two approaches that are of the most interest are (i) and (ii). Therefore we shall first say a few words on the other two methods and will then describe in more detail the first two. Method (iii) consists in regarding two critical points x, y so that |x| = |y| + 1 as an attractor-repellor pair and to apply the general Conley theory of Morse decompositions to this situation (Salamon, 1985). Method (iv) has been introduced by Witten (1982) and is based on a deformation of the differential of the de Rham complex which provides a new differential with respect to which the harmonic forms are in bijection with the critical points of f . Method (i) has been probably folklore for a long time but it first appeared explicitly in Witten’s paper. It is based on noticing that the moduli space MQP admits P

a compactification M Q which is a compact, topological manifold with boundary so that the boundary verifies the formula:  P P R MR × M Q. (1) ∂M Q = R

There are two main ways to prove this formula. One is analytic and regards a flow line from P to Q as a solution of a differential equation x˙ = −∇ f (x) and studies the properties of such solutions (this method is presented in Schwarz, 1993). A second approach is more topological/dynamical in nature as is described in detail ) y by Weber (2004). Clearly, from formula (1) we immediately deduce y ayx az = 0 and hence d2 = 0. Just a little more work is needed to deduce from here the second property. Method (ii) was the one best known classically and it is essentially implicit in Milnor’s h-cobordism book (Milnor, 1965a). It is based on the observation that ayx can be viewed as follows. First, to simplify slightly the argument assume that the only critical points in f −1 ([ f (y), f (x)]) are x and y. It is well known that for
 a ∈ f (y), f (x) , there exists a deformation retract r: M(a) = f −1 (−∞, a] → M( f (y) − ) ∪φy D|y| = M where the attaching map φy : S uf (y)− (y) → M( f (y) − )

(2)

is just the inclusion and  is small. This deformation retract follows the flow till
 reaching of U W u (y) ∪ M( f (y) − ) where U(W u (y) is a tubular neighborhood of W u (y) so that the flow is transverse to its boundary and then collapses this neighborhood to W u (y) by the canonical projection. Clearly, applying this remark to each critical point of f provides a CW-complex of the same homotopy type as that of M and with one cell x¯ for each element of x ∈ Crit( f ). To this cellular decomposition we may associate a cellular complex (C ( f ), d ) with the property ) that d x¯ = kyx¯¯ y¯ where kyx¯¯ is, by definition, the degree of the composition: φx

r

u

− M → − M /M( f (y) − ) ψyx : S au (x) −→ M(a) →

(3)

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with φ x : S au (x) → M(a) again the inclusion and where the last map, u, is the projection onto the respective topological quotient space (which is homeomorphic to the sphere S |y| ). Notice now that Myx ⊂ S au (x) is a finite union of points say P1 , . . . , Pk . Imagine a small disk Di ⊂ S au (x) around Pi . The key (but geometrically clear) remark is that the composition of the flow γ together with the retraction r transports Di (if it is chosen sufficiently small) homeomorphically onto a neighborhood of y inside W u (y). Therefore, the degree of deg(ψyx ) = ayx and thus d = d which shows that d is a differential and that the homology it computes agrees with the homology of M. As we shall see further, the points of view reflected in the approaches at (i) and (ii) lead to interesting applications which go much beyond “classical” Morse theory. Method (iv), while striking and inspiring appears for now not to have been exploited efficiently. 2.1. CONNECTING MANIFOLDS

One way to look to the Morse complex is by viewing the coefficients ayx of the differential as a measure of the 0-dimensional manifold Myx . The question we discuss here is in what way we can measure algebraically the similar higher dimensional moduli spaces. This is clearly a significant issue because, obviously, only a very superficial part of the dynamics of the negative gradient flow of f is encoded in the 0-dimensional moduli spaces of connecting flow lines. As a matter of terminology, the space MQP when viewed inside the unstable sphere S au (P) (with f (P) − a positive and very small) is also called (Franks, 1979) the connecting manifold of P and Q. It was mentioned above that, in general, a connecting manifold MQP is not closed. However, if the critical points P and Q are consecutive in the sense that R  ∅, then M P is closed. there does not exist a critical point R so that MRP × MQ Q 2.1.1. Framed cobordism classes An important remark of John Franks (1979) is that connecting manifolds are canonically framed. First recall that a framed manifold V is a submanifold V → S n which has a trivial normal bundle together with a trivialization of this bundle. Two such trivializations are equivalent (and will generally be identified) if they are restrictions of a trivialization of the normal bundle of V × [0, 1] inside S n × [0, 1]. We also recall the Thom – Pontryagin construction in this context (Milnor, 1965b). Assuming V → S n is framed we define a map φV : S n → S codim(V) as follows: consider a tubular neighborhood U(V) of V, use the framing to define a homeomorphism ψ: U(V) → Dk × V where Dk is the closed disk of dimension k =

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p1

codim(V), consider the composition ψ : U(V) − → Dk × V −→ Dk → Dk /S k−1 = S k and define φV by extending ψ outside U(V) by sending each x ∈ S n \U(V) to the base point in Dk /S k−1 = S k . The homotopy class of this map is the same if two framings are equivalent. It is easy to see that two framed manifolds (of the same dimension) are cobordant iff their associated Thom maps are homotopic. We return now to Franks’ remark and notice that the manifolds MQP are framed. First, we make the convention to view MQP as a submanifold of the unstable sphere of P, S au (P) (the other choice would have been to use S s (Q) as ambient manifold). Notice that we have MQP = S au (P) ∩ W s (Q) and this intersection is transversal. Clearly, as W s (Q) is homeomorphic to a disk, its normal bundle in M is trivial and any two trivializations of this bundle are equivalent. This implies that the normal bundle of MQP → S au (P) is also trivial and a trivialization of the normal bundle of W s (Q) provides a trivialization of this bundle which is unique up to equivalence. Recall that if P and Q are consecutive critical points, then MQP is closed. As we have seen that it is also framed we may associate to it a framed cobordism class |Q| *P ∈ π M |P|−1 (S ). Q

Moreover, it is easy to see that the function f may be perturbed without modifying the dynamics of the the negative gradient flow so that the cell attachments corresponding to the critical points Q and P are in succession. Therefore the map ψPQ : S |P|−1 → S |Q| defined as in formula (3) is still defined. The main result of Franks in Franks (1979) is: THEOREM 2.1 (Franks, 1979). Assume P and Q are consecutive critical points *P coincides with the homotopy class of ψP . of f . Up to sign M Q Q The idea of proof of this result is quite simple. All that is required is to make even more precise the constructions used in the approach (ii) used to show d2 = 0 for the Morse complex. For this we fix for W s (Q) a normal framing o which is invariant by translation along the flow γ and which at Q ∈ W s (Q) is given by a basis e of T Q W u (Q) (this is possible because W u (Q) and W s (Q) intersect
 transversally at Q). We also fix the tubular neighborhood U W u (Q) so that the
 projection r : U W u (Q) → W u (Q) has the property that (r )−1 (Q) = W s (Q) ∩ U(W u (Q)) and, for any point y ∈ (r )−1 (Q), we have (r )∗ (oy ) = e. Moreover, we may assume that the normal bundle of MQP in S au (P) is just the restriction of the

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normal bundle of W s (Q) (in fact, the two are, in general, only isomorphic and not equal but this is just a minor issue). Now, follow what happens with the framing of MQP along the composition u◦r.
 For this we write r = r ◦ r where r follows the flow till reaching U W u (Q) . Now pick a point in x ∈ MQP together with its normal frame o x at x. After applying
 r , the pair (x, o x ) is taken to a pair (x , o x ) with x ∈ ∂ (r )−1 (Q) . Applying now r , the image of (x , o x ) is (Q, e). Take now V a tubular neighborhood of MQP → S au (P) together with an identification V ≈ D|Q| × MQP which is provided by the framing o. The argument above implies that if the constant  used to construct the map u: M( f (Q) − ) ∪ W u (Q) = M → M /M( f (Q) − ) is very small, then the composition u ◦ r ◦ r coincides with the relevant Thom – Pontryagin map. 2.1.2. Framed bordism classes and Hopf invariants It is natural to wonder whether, besides their framing, there are some other properties of the connecting manifolds which can be detected algebraically. A useful point of view in this respect turns out to be the following: imagine the elements of MQP as path or loops on M. The fact that they are paths is obvious (we parametrize them by the value of − f ; the negative sign gives the flow lines the orientation coherent with the negative gradient) but they can be transformed into loops very easily. Indeed, fix a simple path in M which joins all the critical points  which has the of f and contract this to a point thus obtaining a quotient space M  same homotopy type as M. Let q: M → M be the quotient map. We denote by ΩM the space of based loops on M and keep the notation q for the induced map  This discussion shows that there are continuous maps ΩM → Ω M.  jPQ : MQP → Ω M. These maps have first been defined and used in Cornea (2002a) and they have some interesting properties. For example, given such a map jPQ and assuming that P and Q are consecutive it is natural to ask whether the homology class [MQP ] ∈ H|P|−|Q|−1 (ΩM; Z) is computable (here [MQP ] is the image by jPQ of the fundamental class of MQP ). We shall see that quite a bit more is indeed possible: the full framed bordism class associated to jPQ and to the canonical framing on MQP can be expressed as a relative Hopf invariant. To explain this result first recall that if V → S n is framed and l: V → X is a continuous map with V a closed manifold we may construct a richer Thom – Pontryagin map as follows. We again consider a tubular neighborhood U(V) of V in S n together with an identification U(V) ≈ Dk × V where k = codim(V) which is provided by the framing. We now define a map j

id×l φ¯ V : U(V) → − Dk × V −−−→ Dk × X → (Dk × X)/(S k−1 × X)

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where j: U(V) = Dk × V → V is the projection and the last map is just the quo
 tient map (which identifies S k−1 × X to the base point). Notice that φ¯ V ∂U(V) ⊂ S k−1 × X. Therefore, we may extend the definition above to a map φ¯ V : S n → (Dk × X)/(S k−1 × X) by sending all the points in the complement of U(V) to the base point. It is well known (and a simple exercise of elementary homotopy theory) that there exists a (canonical) homotopy equivalence (Dk × X)/(S k−1 × X) Σk (X + ) where ΣX is the (reduced) suspension of X, Σi is the suspension iterated i-times and X + is the space X with an added disjoint point (notice also that Σk (X + ) = Σk X ∨ S k where ∨ denotes the wedge or the one point union of spaces). This allows us to view the map φ¯ V as a map with values in Σk (X + ). The framed bordism class of V is simply
 the homotopy class [φ¯ V ] ∈ πn Σk (X + ) . This is independent of the various choices made in the construction. Two pairs of data (framings included) (V, l) and (V , l ) admit an extension to a manifold W ⊂ S n × [0, 1] with ∂W = V × {0} ∪ V × {1} iff φ¯ V φ¯ V . Notice also that to an element α ∈ πn (Σk X + ) we may associate a homology class [α] ∈ Hn−k (X) obtained by applying the Hurewicz homomorphism, desuspending k-times and projecting on the H∗ (X) term in H∗ (X + ). Returning now to our connecting manifolds MQP we again focus on the case when P and Q are consecutive. The map jPQ together with the canonical framing 
provide a homotopy class {MQP } ∈ π|P|−1 Σ|Q| (ΩM + ) (to simplify notation we have  with M here — the two are homotopy equivalent). replaced M As indicated above, it turns out that this class can be computed in terms of a relative Hopf invariant. We shall now discuss how this invariant is defined. β

α

Assume that S q−1 − → X0 → X and S p−1 → − X → X are two successive cell attachments and that X is a subspace of some larger space X. In particular, X = X0 ∪α Dq , X = X ∪β D p . Let S ⊂ Dq be the q − 1-sphere of radius 12 . There is an important map called the coaction associated to α which is defined by the composition ∇: X → X /S ≈ S q ∨ X where the first map identifies all the points of S to a single one and the second is a homeomorphism (in practice it is convenient to also assume that all the maps and spaces involved are pointed and in that case we view Dq as the reduced cone over S q−1 ). We consider the composition β

∇

id∨ j

− X − → S q ∨ X −−−→ S q ∨ X → S q ∨ X ψ(β, α): S p−1 → and notice that if p2 : S q ∨ X → X is the projection on the second factor, then the composition p2 ◦ψ(β, α) is null-homotopic. This is due to the fact that this compoβ

sition is homotopic to S p−1 → − X → X → X which is clearly null-homotopic.

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We now consider the map p2 : S q ∨ X → X. It is well known in homotopy theory that any map may be transformed into a fibration. In our case this comes down to considering the free path fibration  →X t: PX  is the set of all continuous path in X parametrized by [0, 1], t(γ) = γ(0). where PX  of the resulting We take the pull-back of this fibration over p2 . The total space E q fibration has the same homotopy type as S ∨ X and it is endowed with a canonical projection → X p: ˜ E which replaces p2 , p(z, ˜ γ) = γ(1). It is an exercise in homotopy theory to see
 that the fibre of the fibration  p is homotopic to Σq (ΩX)+ and that, moreover, the inclusion of this fibre in the total space is injective in homotopy. As the composition p2 ◦ ψ(β, α) is homotopically trivial, the homotopy exact sequence of the
  → X implies that ψ(β, α) admits a lift to ψ(α, ¯ β): S p−1 → Σq (ΩX)+ fibration E whose class does not depend on the choice of lift. We let H(α, β) ∈  homotopy
 q + π p−1 Σ (ΩX) be equal to this homotopy class and we call it the relative Hopf invariant associated to the attaching maps α and β (for a discussion of the relations between this Hopf invariants and other variants see Chapters 6 and 7 in Cornea et al., 2003). To return to Morse theory, recall from (2) that passing through the two consecutive critical points Q and P leads to two successive attaching maps φQ : S |Q|−1 → M( f (Q) − ) and φP : S |P|−1 → M( f (P) − ) (we assume again — as we may — that the set f −1 ([ f (Q), f (P)]) does not contain any other critical points besides P and Q). Moreover, as we know the inclusion M = M( f (Q) − ) ∪ W u (Q) → M( f (P) − ) is a homotopy equivalence. Therefore, the construction above can be applied to φQ and φP and it leads to a relative Hopf invariant 
H(P, Q) ∈ π|P|−1 Σ|Q| (ΩM + ) . With these constructions our statement is: THEOREM 2.2 (Cornea, 2002a; 2002b). The homotopy class H(P, Q) coincides (up to sign) with the bordism class {MQP }. In particular, the homology class [MQP ] equals (up to sign and desuspension) the Hurewicz image of H(P, Q). The proof of this result can be found in Cornea (2002b) (a variant proved by a slightly different method appears in Cornea, 2002a). The proof is considerably more complicated than the proof of Theorem 2.1 so we will only present a rough justification here. To simplify notation we let M0 = M( f (Q) − ). Let M1 =
 M( f (Q) − ) ∪ U W u (Q) . Recall, that the inclusions M → M1 → M( f (P) − ) are homotopy equivalences.

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Let P: ΩM → PM → M be the path-loop fibration (of total space the based paths on M and of fibre the space of based loops on M). We denote by E0 the total space of the pull-back of the fibration P over the inclusion M0 ⊂ M. Similarly, we let E1 be the total space of the pull-back of P over the inclusion M1 → M. The key remark is that the attaching map φP : S u (P) → M1 admits a natural lift 'P : S u (P) → E1 . Indeed, we assume here that all the critical points to a map φ are identified to the base point. The space E1 consists of the based paths in M that end at points in M1 . But each element of the image of φP corresponds to precisely such a path which is explicitly given by the corresponding flow line (we need to use here Moore paths and loops which are paths parametrized by arbitrary intervals [0, a] and not only the interval [0, 1]). Consider the inclusion E0 → E1 . It is not difficult to see that the quotient topological space E1 /E0 admits a canonical homotopy equivalence η : E1 /E0 → Σ|Q| (ΩM + ). Therefore, we may consider the 'P . It is possible to show that this map η is homotopic to composition η = η ◦ φ 'P to M P coincides with H(P, Q). At the same time, we see that the restriction of φ Q 'P is homotopic jPQ . Moreover, by making explicit η it is also possible to see that φ P to the Thom – Pontryagin map associated to MQ . 2.1.3. Some topological applications We now describe a couple of topological applications of Theorem 2.2. The idea behind both of them is quite simple: the function − f is also a Morse function and the critical points Q, P are consecutive critical points for − f . Therefore, the connecting manifold MPQ is well-defined as well as its associated bordism homotopy class {MPQ }. Clearly, the underlying space for both MQP and MPQ is the same. The P map jQ P is different from jQ just by reversing the direction of the loops. The two relevant framings may also be different. The relation between them is somewhat less straightforward but it still may be understood by considering M embedded inside a high-dimensional Euclidean space and taking into account the twisting induced by the stable normal bundle. In all cases, this establishes a relationship between the two Hopf invariants H(P, Q) and H(Q, P).

A. The first application (Cornea, 2002a) concerns the construction of examples of nonsmoothable, simply-connected, Poincar´e duality spaces. The idea is as follows: we construct Poincar´e duality spaces which have a simple CW-decomposition and with the property that for certain two successively attached cells e, f the resulting Hopf invariant H and the Hopf invariant H associated to the dual cells f , e are not related in the way described above. If the respective Poincar´e duality space is smoothable, then the given cell decomposition can be viewed as associated to an appropriate Morse function and this leads to a contradiction. The obstructions to smoothability constructed in this way are obstructions to the lifting of the Spivak normal bundle to BO. This is an obstruction theory problem but one

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which can be very difficult to solve directly in the presence of many cells. Thus, this approach is quite efficient to construct examples. B. The second application (Cornea, 2004) concerns the detection of obstructions to the embedding of CW-complexes in Euclidean spaces in low codimension. The argument in this case goes roughly as follows. If the CW-complex X embeds in S n , then we may consider a neighborhood U(X) of X which is a smooth manifold with boundary. We consider a smooth Morse function f : U(X) → R which is constant, maximal and regular on the boundary of U(X). If P and Q are two consecutive critical points for this function we obtain that Σk H(P, Q) = Σk H(Q, P) for certain values of k and k which can be estimated explicitly — the main reason for this equality is that the Morse function in question is defined on the sphere so all the questions involving the stable normal bundle become irrelevant. If X admits some reasonably explicit cell-decomposition it is possible to express H(P, Q) as the Hopf invariant H of some successive attachment of two cells e, f and H(Q, P) as Σk H where H is another similar Hopf invariant. The obstructions to embedding appear because the low codimension condition translates to the fact that k + k > k. This can be viewed as an obstruction because it means that after k suspensions the homotopy class of H has to desuspend more than k-times. REMARK 2.3. The applications at A and B are purely of homotopical type. It is natural to expect that the Morse theoretical arguments that were used to establish these statements can be replaced by purely homotopical ones but this has not been done till now.

2.1.4. The Serre spectral sequence Theorem 2.2 provides considerable information on connecting manifolds for pairs of consecutive critical points. However, it does not shed any light on the case of nonconsecutive ones. Clearly, if the critical points are not consecutive the respective connecting manifold is not closed and thus no bordism or cobordism class can be directly associated to it. However, after compactification, the boundary of this connecting manifold has a special structure reflected by equation (1). As we shall see following Barraud and Cornea (2003), this structure is sufficient to construct an algebraic invariant which provides an efficient “measure” of all connecting manifolds. This construction is based on the fact that the maps jPQ : MQP → ΩM are compatible with compactification and with the formula (1) in the following sense. Recall that here ΩM are the based Moore loops on M (these are loops parametrized by intervals [0, a]), the critical points of f have been identified to

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a single point and, moreover, in the definition of jPQ we use the parametrization of the flow lines by the values of − f . Recall that we have a product given by the concatenation of loops µ: ΩM × ΩM → ΩM. With these notations it is easy to see that we have the following formula: jPQ (u, v) = µ( jRP (u), jRQ (v)) P

R

(4)

P

where (u, v) ∈ M R × M Q ⊂ ∂M Q . We proceed with our construction. Let C∗ (X) be the (reduced) cubical complex of X with coefficients in Z/2. Notice that there is a natural map Ck (X) ⊗ Ck (Y) → Ck+k (X × Y). x

A family of cubical chain syx ∈ C|x|−|y|−1| (M y ), x, y ∈ Crit( f ) is called a representing chain system for the moduli spaces Myx if for each pair of critical points x, z we have: (i)  y dszx = syx ⊗ sz y

(ii) szx represents the fundamental class in H|x|−|z|−1 (Mzx , ∂Mzx ). It is easy to show by induction on the index difference |x| − |z| that such representing chain systems exist. We now fix such a representing chain system {syx } and we define ayx ∈ C|x|−|y|−1 (ΩM) by ayx = ( jyx )∗ (syx ). Notice that this definition extends the definition of these coefficients in the usual Morse case when |x| − |y| − 1 = 0. We have a product map C∗ (µ)

·: Ck (ΩM) ⊗ Ck (ΩM) → Ck+k (ΩM × ΩM) −−−−→ Ck+k (ΩM) which makes C∗ (ΩM) into a differential ring. The discussion above shows that inside this ring we have the formula  dazx = azx · azy . y

An elegant way to rephrase this formula is to group these coefficients in a matrix A = (ayx ) and then we have (5) dA = A2 .

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We now define a new chain complex C( f ) associated to f by  C( f ) = (C∗ (ΩM) ⊗ Z/2Crit( f ), d), dx = ayx ⊗ y.

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(6)

y

We shall call this complex the extended Morse complex of f . Here, C∗ (ΩM) ⊗ Z/2Crit( f ) is viewed as a graded C∗ (ΩM)-module and d respects this structure in the sense that it verifies d(a ⊗ x) = (da) ⊗ x + a(dx) (the grading on Crit( f ) is given, as before, by the Morse index). Choosing orientations on all the stable manifolds of all the critical points induces a co-orientation on all the unstable manifolds, and hence an orientations on the intersections W u (P) ∩ W s (Q) and finally on all the moduli spaces MQP : we may then use Z-coefficients for this complex as well as, of course, for the classical Morse complex. In this case appropriate signs appear in the formulas above. Clearly, d2 = 0 due to (5). By definition, the coefficients ayx represent the moduli spaces Myx . However, these coefficients are not invariant with respect to the choices made in their construction. Therefore, it is remarkable that there is a natural construction which extracts from this complex a useful algebraic invariant which is not just the homology of the complex — as it happens, this homology is not too interesting as it coincides with that of a point. Consider the obvious differential filtration which is defined on this complex by F k C( f ) = C∗ (ΩM) ⊗ Z/2x ∈ Crit( f ) : ind f (x) ≤ k. Denote the associated spectral sequence by E( f ) = (E rp,q ( f ), dr ). THEOREM 2.4 (Barraud and Cornea, 2003). When M is simply connected and if r ≥ 2 the spectral sequence E( f ) coincides with the Serre spectral sequence of the path-loop fibration P: ΩM → PM → M. REMARK 2.5. (a) A similar result can be established even in the absence of the simple-connectivity condition which has been assumed here to avoid some technical complications. (b) The Serre spectral sequence of the path-loop fibration of a space X contains considerable information on the homotopy type of the space. In particular, there are spaces with the same cohomology and cup-product but which may be distinguished by their respective Serre spectral sequences.

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To outline the proof of the theorem we start by recalling the construction of the Serre spectral sequence in the form which will be of use here. We shall assume here that the Morse function f is self-indexed (in the sense that for each critical point x we have ind f (x) = k ⇒ f (x) = k) and that it has a single minimum
 denoted by m. Let Mk = f −1 (−∞, k + ] . We have  Dky Mk = Mk−1 φy

where the union is taken over all the critical points y ∈ Critk ( f ) and φy : S u (y) → Mk−1 are the respective attaching maps. Denote by Ek the total space of the fibration induced by pull-back over the inclusion Mk → M from the fibration P.
 Consider the filtration of C∗ (PM) given by F k P = Im C∗ (Ek ) → C∗ (PM) . The spectral sequence induced by this filtration is invariant after the second page and is precisely the Serre spectral sequence (this spectral sequence may be constructed as above but by using an arbitrary skeletal filtration {Xk } of a space X which has the same homotopy type as that of M; in our case the particular filtration given by the sets Mk is a natural choice). For further use, we also notice that there is an obvious action of ΩM on PM and this action induces one on each Ek . Therefore, we may view C∗ (Ek ) as a C∗ (ΩM)-module. The first step in proving the theorem is to consider a certain compactification of the unstable manifolds of the critical points of f . Recall that f is self-indexed and that m is the unique minimum critical point of f . Fix x ∈ Crit( f ) and define x the following equivalence relation on the set M m × [0, f (x)]: (a, t) ∼ (a , t )

iff

t = t and a(−τ) = a (−τ) ∀τ ≥ t.

x

Here the elements of M m are viewed as paths in M parametrized by the value of
 − f (so that f a(−τ) = τ).  Denote by W(x) the resulting quotient topological space. Notice that, if x u y ∈ W (x), then there exists some a ∈ M m so that y is on the (possibly broken) flow line represented by a. Or, in other words, so that a(− f (y)) = y. This path a  might not be unique. Indeed, inside W(x) there is precisely one equivalence class
 [a, f (y)] (with a − f (y) = y) for each (possibly) broken flow line joining x to y. Clearly, if y ∈ W u (x), then there is just one such flow line and so the natural surjection  π: W(x) → W u (x), π([a, t]) = a(−t)  as a is a homeomorphism when restricted to π−1 (W u (x)). Thus we may view W(x) u u special compactification of W (x) or as a desingularization of W (x). It is not difficult to believe (but harder to show and will not be proved here; see Barraud and

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 Cornea, 2003) that W(x) is a topological manifold with boundary and moreover    ∂W(x) = Myx × W(y). (7) y

We continue the proof of the Theorem 2.4 with the remark that there are obvious maps  h x : W(x) → PM which associate to [a, t] the path in M which follows a from x to a(−t). These maps and the maps jyx are compatible with formula (7) in the sense that h x (a , [a , t]) = jyx (a ) · hy ([a , t])

(8)

 and · represents the action of ΩM on PM. where (a , [a , t]) ∈ Myx × W(y) Now, we may obviously rewrite (7) as:  x   M y × W(y). ∂W(x) = y

Given the representing chain system {syx } it is easy to construct an associated
   representing chain system for W(x). This is a system of chains v(x) ∈ C|x| W(x)
   and we have the so that v(x) represents the fundamental class of C|x| W(x), ∂W(x) formula  dv(x) = syx ⊗ v(y). y

Finally, we define a C∗ (ΩM)-module chain map α: C( f ) → C∗ (PM) by α(x) = (h x )∗ (v(x)). The formulas above show that we have
  x
  d (h x )∗ v(x) = ay · (hy )∗ v(y) y

and so α is a chain map. It is clear that the map α is filtration preserving and it is not difficult to see that it induces an isomorphism at the E 2 level of the induced spectral sequences and this concludes the proof of Theorem 2.4. REMARK 2.6.

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 (a) Another important but immediate property of W(x) is that it is a contractible x space. Indeed, all the points in M m × { f (x)} are in the same equivalence  class. Moreover, each point [a, t] ∈ W(x) has the property that it is related  by the path [a, τ], τ ∈ [t, f (x)] to ∗ = [a, f (x)]. The contraction of W(x)   to ∗ is obtained by deforming W(x) along these paths. Given that W(x) is a contractible topological manifold with boundary, it is natural to suspect  that W(x) is homeomorphic to a disk. This is indeed the case as is shown in Barraud and Cornea (2003) and is an interesting fact in itself because it implies that the union of the unstable manifolds of a self-indexed Morse – Smale function gives a CW-decomposition of M. The attaching map of the   cell W(x) is simply the restriction of π to ∂W(x). (b) The Serre spectral sequence result above and the bordism result in Theorem 2.2 are obviously related via the central role of the maps jPQ . There is also a more explicit relation. Indeed, (a stable version of) the Hopf invariants appearing in Theorem 2.2 can be interpreted as differentials in the Atyiah – Hirzebruch – Serre spectral sequence of the path-loop fibration with coefficients in the stable homotopy of ΩM. Moreover, the relation (1) can be understood as also keeping track of the framings. This leads to a type of extended Morse complex in which the coefficients of the differential are stable Hopf invariants (Cornea, 2002a). All of this strongly suggests that the construction of the complex C( f ) can be enriched so as to include the framings of the connecting manifolds and, by the same method as above, that the whole Atyiah – Hirzebruch – Serre spectral sequence should be recovered from this construction. (c) Another interesting question, open even for consecutive critical points P, Q, is whether there are some additional constraints on the topology of the connecting manifolds MQP besides those imposed by Theorem 2.2. (d) Yet another open question is how this machinery can be adapted to the Morse – Bott situation and how it can be extended to general Morse – Smale flows (not only gradientlike ones). (e) It is natural to wonder what is the richest level of information that one can extract out of the moduli spaces of Morse flow lines. At a naive level, the union of all the points situated on the flow lines of f is precisely the whole underlying manifold M so we expect that there should exist some assembly process producing the manifold M out of these moduli spaces. Such a machine has been constructed by Cohen et al. (1995b; 1995a). They show that one can form a category out of the moduli spaces of connecting trajectories and that the classifying space of this category is of the homeomorphism type of the underlying manifold. In their construction an essential point is that the gluing of flow lines is associative. This approach is quite different from

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the techniques above and does not imply the results concerning the extended Morse complex or the Hopf invariants that we have presented. The two points of view are, essentially, complementary. To end this section it is useful to make explicit a relation between Theorems 2.4 and 2.2 (we assume as above that M is simply-connected). PROPOSITION 2.7. Assume that there are q, p ∈ N so that in the Serre spectral 2 = 0 for q < k < p and there sequence of the path loop fibration of M we have Ek,s is an element a ∈ E 2p,0 so that d p−q a  0, then any Morse-Smale function on M has a pair of consecutive critical points P, Q of indexes at least q and at most p so that the homology class [MQP ] ∈ H|P|−|Q|−1 (ΩM)  0. Clearly, Theorem 2.4 directly implies that, even without any restriction on E 2 , if we have dr a  0 with a �