Morse Homology (Progress in Mathematics 111)

'[JT2 0

'l/J

0


--+

0

'l/J

0

--+

Proof of proposition 2.24: Owing to the given countable trivialization of E we are able to assume without loss of generality that E is a trivial bundle. This is to say that we have a smooth map


be a be a is a

with respect to a Banach space IE, such that 0 E IE is a regular value and the maps

nu

that

countable trivialization {U, 'l/Ju}. Thus,

(2.22)

lEu and

= {m

(2.23)

E M

E x F

= ker

for all z

= (g, m)

E Z ,

is a Fredholm map endowed with the same index as

ker D1f(z)

= TzZ n TmM = ker D 2
,

of finite-dimensional vector spaces with the assumption that D 2
Lemma 2.25 Let

--+

ker D
1f:Z--+G,

We call the main result of this section the transversality result. The reader may notice that transversality in this sense and concerning the trajectory spaces is equivalent to the Morse-Smale condition, which is also defined in a way that describes a kind of transversality. The essential step of the proof is based on the following technical lemma, which can be proved easily by straightforward methods on Banach spaces.

Let
=

Now the main step of the proof is to conclude that the restriction of the projection map to this Banach manifold Z,

Given a generic parameter set 1:; C G, the section

=
TzZ

as follows via local coordinate charts from Lemma 2.25 (!) and from the implicit function theorem.

Another way of putting this would be to say:


Z =
is a smooth Banach manifold with associated tangent spaces given by

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

Then there is a Q-set 1:; C G, such that the set Zg =
43

TRANSVERSALITY

(2.25)

.

DI
------. f--->

E / R( D 2
[D I

Having in mind the identities

G be a Fredholm

v=D1f'(v,w)

,

and

D I
for each pair (v, w) E TzZ c TgG x TmM, we immediately deduce the isomor­ phism property of DI

such that H is a closed subspace of Ex F. ,;~~ jD'

...

r--------------------­

--------------------------------------1-­

II

CHAPTER 2. THE TRAJECTORY SPACES'"

I

44

cokernelof D 2 cI>(z) has a finite dimension, D 1 cI> appears to be an isomorphism between finite-dimensional cokernels, (2.26)

D 1 cI>: coker (D1l"(z)) ~ coker D 2 cI>(z)

Hence, the equations (2.24) and (2.26) yield the Fredholm property of the projection map 1l". Finally, we are able to apply the well-known theorem of Sard-Smale to this smooth Fredholm map between Banach manifolds. It leads to a 9-set ~ C G Of regular values of 1l".

I

Now let b E ~. If cI>(b, m) does not vanish for any m from M, 0 is trivially a regular value of cI>b. Therefore, let us consider an m E M satisfying cI>(b, m) = O. We then have to show that the operator (2.27)

DcI>b(m) = D 2 cI>(b, m) : TmM

~E

is onto. Let 'Y E E be given arbitrarily. Due to the regularity of b with respect to cI>, there is a pair (0:,/3) EnG x TmM satisfying (2.28)

On the other hand, b is also regular with respect to 1l" : Z ~ G, that is, given any 0: E TbG we find an (0:', /3') E T(b,m)Z, such that it holds

norm 1·1 and a covariant derivation. Given a sequence real numbers, we define

Xl'"

This norm

I ,11,

E

TpM}

I Ilsll,

< 00 }

It is easy to verify that (C'(~), 11,11,) is a Banach space if C'(~) is non­ void. An explicit construction of a sufficiently large set of smooth sections with suitably chosen compact support, which the reader can find in [F3], gives rise to the following lemma. (fn)nEN

such that the Banach space C'(~)

Remark Obviously, the following inequality holds:

11,11,

~

II-II",

if 0 < f n ~ f~ for all n E N

In other words, we have a continuous embedding of C" in C'. Summing up, we get the sequence of embeddings C" c~. C' c~. C k c~. CO for all kEN .

(2.31) Subtracting equation (2.30) from (2.28), we find a preimage of'Y under the map D 2 cI>(b, m) of the form

This yields the proof.

Xi

of positive

gives rise to the vector space C'(~) = { s E COO(~)

0= DcI>(b, m) . (0:, /3') = D 1 cI>. 0: + D 2 cI> . /3'

'Y = D 2 cI>(b, m) . (/3

(fn)nEN

Ilsll, = Ek~O fk sUPM lV'ksl for all s E COO(~) , lV'ksl (P) = max { IV' V'xks(P)llllxtli = ... = Ilxk 11= 1,

0:' = D1l"(b,m)· (0:',/3') = 0: .

Hence, (0:,/3') belongs to the kernel of DcI>(b,m), Le. (2.30)

Definition 2.26 Let ~ be a smooth vector bundle on M endowed with a bundle

Lemma 2.27 There is a sequence is dense in L 2 (~).

'Y = DcI>(b, m) . (0:, /3) = D 1 cI>. 0: + D 2 cI>· /3

(2.29)

45

2.3. TRANSVERSALITY

- /3') .

o

It is the aim of this section to obtain a generic set of smooth Rie­ mannian metrics, each of which satisfies the Morse-Smale condition. Within the given framework it is sufficient to fix an arbitrary metric 90 and to find a generic set of variations with respect to 90. By this we mean a set of smooth sections A E End (TM) which are fibrewise self-adjoint with respect to 90, positively definite and which differ from the section of identities by a suffi­ ciently small amount. We then vary V' f in terms of A . V' f. In order to be able to apply Proposition 2.24 to our transversality problem, we need a Banach manifold structure upon this parameter set of smooth sections in End (T M).

Definition 2.28 Setting

Endsym,go (TM) {(p, T p) E EgO I T p positively definite w.r.t. (-, -}p = 90(P)}

Ego Tgo

we now define the vector bundle Ego on M. It consists of the endomor­ phisms which are self-adjoint with respect to 90, so that it forms a subbundle of End(T M) ~ M. Additionally, Tgo represents an open neighbourhood of the submanifold nM in EgO C End(TM). Due to the above remark C'(~) ~ CO(~) is a continuous embedding and therefore we obtain

9go

= C'(~o) = {X E C'(Ego ) IX p

as an open neighbourhood of

nE C'(Ego )'

E (Tgo)p f.a. P EM}

CHAPTER 2. THE TRAJECTORY SPACES

46

Referring to the local coordinates and the trivialization of L2(P~:~) we compute the represer.tation of as

Thus, 9go is a suitable Banach manifold of parameters with regard to our regularity analysis. It should be mentioned that each tangent space T x 9go at X E 9go has the form T x 9go = C«Ego ) .

loe

9go x H 1 ,2b*O)

~oexp,~

)

Proposition 2.30 Given any parameter A E 9go '

A = loe(A,') : H 1 ,2b*O) ~ L 2b*TM)

is called regular, if it fulfills the condition:

Given any lR-critical point Xo of h, i.e. =

(A. V'h t

Since C«~o) ~ CO(~o) is a continuous inclusion, loe is also smooth with respect to the parameter variable A. We may now state the decisive proposition, where we denote loe again by <1>.

tt ~ ~R -R for some R ~ 0 ,

V'ht(xo)

~ L 2b*TM)

(A,~)t-+ V't~+e(~)''Y+V'2exp(~)-'

Definition 2.29 A -finite homotopy h a{3 : lR x M ~ lR, i.e. satisfying

r,

:

1

In order to prove the central regularity result already for the time-dependent case which includes the time-independent one as a special case6 we still need a further restriction on the homotopy h a {3 :

h(t , .) _- { f{3,

47

2.3. TRANSVERSALITY

is a Fredholm map with the constant index J.L(x) - J.L(y). regular homotopy, 0 E L 2 is a regular value of <1>.

0 for all t E lR , ~x~TM

the consequently well-defined operator from

Moreover, if ht is a

Proof. a) The Fredholm result is an immediate conclusion from Lemma 2.19 and the investigations in Section 2.2.3, because

a + H 2h (xo) : H 1 ,2(x TM) ~ L 2(x TM) t o o

at

for all A E 9go

A E Fredb*TM,'Y*V')

is onto.

b) The focal regularity result is now obtained as follows: 2 The In this framework, H h t (xo) E End(TxoM) denotes the well-defined (f) Hes­ differential of has the form sian of h t at xo. D(a,~) . (B, 17) = D 1 (A,~) . B + D2<1>(A,~) '17 Generally, this regularity condition is not fulfilled. But if rand f{3 are func­ tions with isolated critical points, we can always find a regular, finite homotopy between them. This is deduced by the following argument. Let us start from a non-regular lR-critical point Xo of h a {3 by assumption. We then choose a func­ tion k E Cgo (lR x M, lR) with suitably small support and satisfying dk t (xo) i- 0 for some t E lR. Let us now replace the homotopy h a{3 by h'a{3 = h a{3 + k. This replacement can be iterated for all at most countably many non-regular lR-critical points within a compact interval of time. We thus obtain a regular, finite homotopy. In particular, h a {3 is a regular homotopy if it is trivial, that is to say if h t = for all t E R

Here, D2<1>(A,~) E ~,.TM is a Fredholm operator of the already well-known type. For sake of computational plainness we choose the short notations

2

d = Dl(A,~)· BE L b*TM) ,

d(t) = V'2 exp(~)-l(t) .

BV'ht (

9go

X p~,2 ~

(A, 'Y)

,y

t-+

'Y +

L 2(pl,2) x,y

A.

J1 + kt/ 1

6h t

= 0 for all t E

-

AV'ht(AV'ht , BV'ht ))

2

( 1 + 1 kt 1

.

I

A V' ht 1

2 )

~

together with

Let us now study the following map: :

2

(BV'h t IAV'h t 1

3

r



+ Ikt l2

V' gOh t

2 .

I

A . V' go h t 1

2

IR .

·'V!'< . i-"

.I·

oexp,~) (t)

a(t)

(lktl2

x(t)

(V'h t 0 exp,~) (t)

y(t)

((AV'h t ) oexp,~) (t)

B(t)

(B 0

exp,~)

(t) .

oexp,~

)

(t)

CHAPTER 2. THE TRAJECTORY SPACES

48

2.3.

Now let (A,~) E 9go X H 1 ,2 satisfy ell(A,~) = O. Then ~ is smooth, too. Since due to a) R(D2ell(A,~)) is a closed subspace of £2 with finite codimension, the same is true for R(Dell). Thus, let us start from any vector c E £2 within the finite-dimensional subspace of £2 satisfying (Dell. (B,17) ,

(2.32) for all

c) £2

=

TRANSVERSALITY

Since

0

49

is positive and P ha.s a non-negative spectrum, it holds that

(n + oP) C i= 0 Therefore, there is a z

(Bx, z)

(2.38)

0

i= 0

satisfying

=

0

for all B E Endsym(TpM)

A simple computation shows that the space of symmetric endomorphisms com­ plies with the property:

(B,17) E C'(E go ) x H 1 ,2(J*TM)

It remains to show that c(t) vanishes for all t E R The relation (2.32) leads

For any pair v,

in particular to

W

i= 0

there exists a symmetric B with (Bv, w)

i= 0

.

Thus (2.38) gives rise to the conclusion (2.33)

( D 2ell '17,

c) £2

= 0

for all 17 E H 1 ,2

.

As we already analysed in the Fredholm section, due to the smoothness ~ E Coo, this implies that c can be expressed a.s the solution of some non-autonomous linear differential equation

= (\lh t

0

expoy ~)(t)

=0

for all t E IR .

However, due to the identity ell ( A,~) = 0, it follows that s = expoy ~ must be a constant trajectory. This leads to a contradiction, a.s we a.ssumed h t to be a regular homotopy thus implying the surjectivity of D 2ell. 0

c(t) = X(t) . c(t)

(2.34)

2.3.2

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

(2.35)

x(t)

(2.39)

(D 1 ell(A,~) . B , c) £2

= 0

c E COO(i*TM) with c(t)

Combining Propositions 2.24 and 2.30 we now are led to the result about transversality.

for all B E C'(Ego ),

i= 0

The Regularity Results

Theorem 1 Given a generic set of smooth metrics, the trajectory spaces Mt,y and M~~: are closed submanifolds ofP;:~ of the finite dimension p,(x) - p,(y).

for all t E R .

Remarks. Due to the above discussion about C' this is generally true for all BE L 2 (Ego ), that is especially for all B with compact support. Choosing B with a suitably small support we may also give a pointwise formulation of (2.35) for a single t E R: Let us fix any arbitrary t E R. Then there exists a

• A priori, we can only speak of local dimensions, because they are in­ duced by the Fredholm indices of the local linearizations of the Fredholm section. These indices are only local analytic invariants which depend continuously on the underlying point of the zero set. The fact that they are equal on each component of this trajectory space follows from the mere dependence of the index formula on the fixed endpoints x and y in this special framework. In fact, if we look at the analogous Fredholm problem in Floer homology, we observe an additional parameter in the index formula, which involves a certain Chern number a.s a strictly local invariant, in general. In contra.st to our situation, the trajectory spaces in Floer homology may have different dimensions for different components.

C E TpM, C i= 0 with p = expoy~(t) ,

satisfying the equation (2.36)

( Bx + o(Bx(y, y) - y(y, Bx)),

c) =

0

for all B E End(TpM), which are symmetric with respect to (-,.) = go (', .) on TpM. Let P be the symmetric operator

• There is a corresponding result for the time-dependent ca.se of homotopy trajectories, when the Riemannian metric is homotoped over a finite in­ terval of time along with h o.13 . Let arbitrary generic metrics a.ssociated to the functions f o. and f13 be given. Then there is a generic set of a.ssociated homotopies gf13, such that M~~: ,g",{3 has the property of a manifold.

I

pz

=

z (y, y) - y (y, z)

r!

I~ II

Then (2.36) may be written a.s (2.37)

((n + oP)Bx, c)

= 0

for all B E Endsym(TpM) .

,' , A

.:;'

f·

CHAPTER 2. THE TRAJECTORY SPACES

50

Finally, we have to study the case of A-parametrized trajectories: Let h~13 and h~13 be finite homotopies of the form

fa, hf13(t,.) = { f13,

t ~-R t~ R ,i

= 0,1 ,

2.4. COMPACTNESS

being a Fredholm operator for each A yields the finiteness of the codimension of R(G). Additionally, it is immediate from the continuity argument that the index of G(A) is constant on the entire interval [0,1]. Hence, we compute the index of G by means of the index formula indG2(0)

and let H a 13 be a A-homotopy satisfying H a 13 : [0,1] x lR x M

r,

\ )_ { H' 0113 (/\, t,' f13, H a13(o , ..) ,

= h 0a13 ,

---+

ind G(A) = indG(O) = 1 + /L(x a ) - /L(Y13)

t~ -R t~ R

H a13(l , . , .)

[0 , 1] (A, 1')

X

p1,2 X ,Y{3

---+

a

= ha13 1

L 2(p1,2* TM) Xo:,Y/3

~ (i'(t) + (V~') 01') (t))tElR

with X a E Crit(fa) and Y13 E Crit(f13). Corresponding to the first remark above, the Riemannian metric determining V may likewise be (A, t)-dependent. The following transversality result holds: Theorem 2 Let h~13 and hr 13 be regular, finite homotopies together with its associated Morse-Smale metrics. Then there is a generic set of A-homotopies H a13 of the above form and a generic set of suitable homotopies of the Rie­ mannian metric, such that (Ga 13 )-l(o) is a (/L(x a ) - /L(Y13) + l)-dimensional submanifold of [0,1] X p;~2,Yi3'

Proof. Apart from the Fredholm property which is needed according to Propo­ sition 2.24 with respect to the fixed local trivialization of the Banach bundle L2(P;~2'~i3TM), the proof of this regularity follows in a way that is analogous to Proposition 2.30. And this Fredholm property is concluded as follows: Given the local trivialization at 1', we consider the linearized map

G

: lR x H 1 ,2(')'*TM)

---+

L 2(')'*TM)

with

o i3

It should be mentioned at this stage that, generally, M{!"'y is a bounded "" 13 manifold and that it is due to the regularity assumption at A = 0,1, that the boundary satisfies the relation

H"'i3

H"'i3

12

8M X""Yi3 = M x ""Yi3 n {O, 1} x PX~,Yi3

u'"

Thus, there are no interior points E M~~:i3 which belong to the parameters A = 0 or A = 1. For a schematic picture of this situation the reader is referred to Figure 4.4 on page 146.

2.4

Compactness

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

i'=-Vf°'Y . Actually, this q~)c-topology appears to convey geometrical significance rather than the H 1 ,2_topology. Due to the deduction from the implicit function theo­ i3 rem the finite-dimensional manifolds M~ Y and M xh : ,x 13 are obviously manifolds without boundary. Thus, in relation to the dimension one is naturally led to the question of compactness. In the case of relative Morse index /L(x) - /L(Y) = 1 this would immediately imply the finiteness of M~,y, and the case of dimensions 1 and 2 would allow an easy classification of the diffeomorphism type. I

(T,~) ~ (G 1(A)' T + G2(A)'~) ,

G2 (A) . ~ = Vt~ + A(A) . ~ , G 1 : [0,1] ---+ L 2(')'*TM), G 2 : [0,1] ---+ I;'Y'™

= dimkerG 2(0) = /L(x a ) - /L(Y13)

and due to the fact that G 2(0) is onto, as

lR

Then we consider the 1-parameter family of sections,

Ga 13

51

both smooth .

First, let us observe that the regularity assumption for the homotopies h~13 and hr13 implies that the operators G2(0) and G 2(1) are onto. Second, G 2(A)

As it will turn out throughout this investigation, the essential ob­ struction to the convergence of subsequences in the H 1 ,2_topology is formed by the geometrical splitting-up into the so-called broken trajectories. Actually, it

===:E2

5;

tEt==.: f,:MitdbEtfn:z . ntt~;:;~;:;=:E ;:;:C:;:;;~=:~; :rr.:;Zejr='~:;~7~~:;:~;~~~5~=:~~~1e~~~~W«4~~h§.

__W_';;­

u.,

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

2.4.1

shall make use of the identification Mt,y ~ WU(x) n WS(y). We consider the smooth evaluation map Pl,2(IR x,y , M)

Eo:

---->M ~ 'Y(O)

'Y

The smoothness of this map follows immediately from the representation by suitable local coordinates, EO,loc:

The Space" of Unparametrized Trajectories

Hi,2(h*D)

---->

~

U(h(O))

c

Th(o)M

~(exPh(1)oEooexPh)(~)=~(0),

The aim of this subsection is to analyse the essential property of the trajec­ tory manifold Mt,y in the time-independent case, namely the symmetry with respect to additive reparametrization of the trajectories. This action is called 'time-shifting'. We shall henceforth use the designation 'time-independent' or 'unparametrized' trajectories for the orbits of this time-shifting operation.

where EO,loc is continuously linear. If we restrict the evaluation map Eo to the trajectory manifold Mt,y, we obtain an embedding

Let 'Y : IR ----> M be a solution of the differential equation 'Y = - \7 f 0'Y. Then, obviously, the same property holds for the shifted curve 'Y.T = 'YT = 'Y(' + T), as we compute

because the differential

(2.40)

Eo : Mt,y

proves injective at each 'Y E

---->

T1(o)M

Mty.

Specifically, let us consider a tangent vector O. This argument analogous to (2.34) in Proposition 2.30, that ~ may be treated as solution of a linear, ordinary differential equation. Thus the identity ~ == 0 is concluded from the uniqueness with respect to the initial value ~(O) = O. As a consequence the evaluation map leads to the diffeomorphism between the manifolds (2.41 )

lR x M~,y

---->

~ E T1Mt,y = ker DF("() satistying ~(O) =

Proposition 2.31 The additive group lR acts smoothly, freely and properly on the manifold Mt,y by

M ,

DEo ("() : T1M~,y DEo ("() . ~ = ~(O)

a at ("(. T) = i{ + T) Moreover, it holds that 'YOR) = ("(. T)(i) for all 'Y E Mt,y, that is the unparametrized set of points 'Y(i) remains unchanged. It is in this sense of point sets that we shall use the expression 'geometrical' behaviour. We make this observation of symmetry precise by the following proposition.

'---+

Eo:

Mt,y

~

'Y

~

WU(x)

n WS(y)

'Y(O)

Mt,y

(T,'Y)~'Y.T='Y(-+T) ,

as it was already mentioned in the introduction. It is obvious that Eo identifies the group action (T, 'Y) ~ 'Y. T with the negative gradient flow,

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

n WS(y)

lR x (WU(x) n WS(y))

---->

WU(x)

(t ,p)

~

Wt(p) .

This gradient flow restricted to WU(x) n W8(y) represents a smooth, free and proper lR-action. Thus the equivalence 'Y •

yields the proof.

T

= (Eo loW

T 0

Eo) ("()

o

'i.

CHAPTER 2. THE TRAJECTORY SPACES

54

't~

Given

In order to describe the orbit space Mt,y = Mt,y/R of the so­ called unparametrized 7 or time-independent trajectories, we define the smooth function (2.42)

f ° Eo : Mt,y ~ R with depb) . ~ = df (-y(0)) . ~(O) = (\7 f(-y(O))

2.4. COMPACTNESS (s,~) E

55

ker DWa(O, 1'), the equation 0= DW a(O,1')' (s,~) = "y.

ep =

S

+~

follows. From "y = -\7f 01' we obtain ~(O) = s· \7f(-y(O)), such that (2.43) implies the identity ~(O) = O. Hence we conclude the injectivity of Dw a (0, 1') together with the surjectivity due to the dimension argument. Finally, we obtain the proof of the assertion from the identity

,~(O))

Definition 2.32 Let f(y) < a < f(x), such that 1'(0) is not critical for any l' E ep-l (a). Then a. is a regular value of ep and

Wa(7, 1')

M~:~ = ep-l (a)

because

is a (J.t(x) - J.t(y) -l)-dimensional submanifold of Mt,y and thus ofP;;;, too.

cPr

=

cPr (w a (0,1')) ,

is a diffeomorphism according to Proposition 2.31.

Summing up this geometrical investigation we obtain

-Ix,y -== MIx,y /Tll> M ~

~

D

wa as a diffeomorphism

MI,a x,y'

From a geometrical point of view the orbit space Mt,y represents exactly this transversal intersection of a surface associated to a regular level of f with the manifold WU(x)nWS(y), which gives an equivalent description of the trajectory space Mt,y as we have already seen. The reader is also referred to [Bo] and [S]. The following proposition yields an appropriate description for the orbit space.

which appears to be quite suitable in order to describe the differentiable struc­ ture on the space of unparametrized trajectories provided by the negative gra­ dient flow of f.

Proposition 2.33 The map

2.4.2 The Compactness Result for

U nparametrized Trajectories

,T,a . 'J:"

•

R x MI,a x,y

~

(7,1')

f---+

MIx,y 1'.7

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

represents a JR-equivariant diffeomorphism with respect to the trivial JR-action on the left and to the action studied in Proposition 2.31 on the right side.

-I _ I / JR Mx,y - Mx,y

Proof. Due to Proposition 2.31 the JR-equivariance

Wa (7

represents a (J.t(x) - J.t(y) -1 )-dimensional manifold. The first observation with respect to this statement is that the function f does not necessarily need to be a Morse function. It is sufficient for f to give rise to a transversal intersection of df and M as a zero section within the cotangent bundle T* M, and we need a metric which is generic with respect to the Morse-Smale condition.

+ CT, 1') = Wa (7, 1'). CT

is obvious. It is similarly straightforward to verify the bijectivity of wa by means of Proposition 2.31 and the uniqueness for solutions of ordinary differ­ ential equations. Thus it is sufficient to prove that

DW(7,1') : JR x T-yM~:~ ~ T-yerM{y is an isomorphism. Let (s,~) E JR x T-yMt:~. Then (2.42) yields the identity (2.43)

df(-y

I"'V

'ji '&~

(0)) . ~(O) = 0 .

·.~..·; i

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

. L:':

:.;1•~:.•. .

:.

,

If we now wish to study the compactness properties of the trajec­ tory manifold associated to f, wehave to introduce further conditions on this function. In fact, we do not really need f to be a Morse function in the sense of coercivity, but to satisfy a more general condition as is fixed in the following definition.

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

CHAPTER 2. THE TRAJECTORY SPACES

56

(P-S)

Every sequence (Xn)nEN C M, such that (If(xn)1 )nEN is bounded and IV' f(xn)1 -+ 0 converges, has a convergent subsequence.

A subset K C Mt,y is called compact up to broken trajectories of order 1/ or up to (1/ -I)-fold broken trajectories exactly if for all (Un)nEN C K: Either (Un)nEN possesses a convergent subsequence, or there are critical points x = Yo, ... , Yi = Y E Crit f, 2 ::::; i ::::;

1/

and connecting trajectories together with associated reparametrization times Vj

E

M£j,YH"

(Tn,j)nEN

C

lR., j = 0, ... , i - I ,

2.4. COMPACTNESS

We are immediately led to the conclusion: Corollary 2.36 If, moreover, it holds that /-l(x) = /-l(Y) finite set of unparametrized trajectories.

u nk "Tnk,j

Cl~c

~

Vj

for some subsequence (nk) kEN. This convergence of trajectories Wj $ W in C~Y(~' M) shall henceforth be called weak convergence, and the weak convergence of unparame­ trized trajectories with respect to suitably provided reparametrization times is denoted by geometrical convergence. The geometrical convergence is sketched in Figure 2.1. Referring to these terms

Yo. Vo

+ 1,

then Mt,y is a

In order to give a proof for the central compactness result we first state an immediate consequence of the Palais-Smale condition. Auxiliary Proposition 2.37 If f satisfies (P-S) , there is a constant depending merely on f, x and Y, such that the set

K;'Y

such that we obtain the convergence

57

= {z E M I f(y)::::;

f(z)::::; f(x), lIV'f(z)11 ::::; f}

f

>0

is relatively compact.

The following lemma shall express the 'geometrical compactness' of the un­ parametrized trajectory spaces in terms of compactness of the parametrized spaces with respect to the C~c-topology.

Lemma 2.38 Every sequence subsequence

(Un)nEN C

Mi,y possesses a weakly convergent

u nk C,C:c -+ v E Coo (TDl ~, M) . Proof Essentially, the strategy for proving this lemma is to execute a transition from the H I ,2_topology of the Hilbert submanifold Mt,y to the C~c-topology. It appears to be of crucial importance that we have a uniform bound on the HI,2-norm for the trajectories with fixed endpoints x, y E erit f. The q~c­ convergence will, in principle, follow iteratively by elliptic regularity. The fact that the trajectories (Un)nEN arise from the negative gradi­ ent flow with respect to the fixed endpoints x and y yields the uniform estimate

YI.~"

t

(2.44) jIUn(T)1 dT= j
VI

Y2 •

t

2

Let d be the Riemannian distance on M. Then the estimate t

d(un(t), un(s))

::::;

~oposition 2.35 If f satisfies the Palais-Smale condition, the manifold

Mt,y is compact up to broken trajectories of order /-l(x) - /-l(Y).

j

IUn(T)1 dT

s

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

for all s::::;t.

s

Holder

::::;

.j It ­ 'I

,U IU.(T)I' s

(2.44)

::::;

J It - sl ..;

f(x) - f(y)

r'-.

dT

y"c;(­

y

CHAPTER 2. THE TRAJECTORY SPACES

58

gives rise to the equicontinuity of (Un)nEN in CO(JR, M). In order to be able to apply the theorem of Arzela-Ascoli we still have to analyse the pointwise convergence. At this stage the uniform L 2 -bound on the derivatives Un as well as the (P-S)-condition appear to be the decisive arguments. Thus let us choose a fixed to E JR. According to Auxiliary Proposition 2.37 we can assume without loss of generality the inequality

!(V'f 0 Un)(to) I ~E Due to the asymptotic decrease

(tn)nEN

c

lim

Itl-oo JR, such that for all n E N:

(2.45) lV'f(un(t n )) I

for all nEN.

59

compact intervals. This amounts to stating that there is a

v E CO(JR, M) with v(t) = lim u nk (t), t E JR, s.t. k-oo CO ([-R,R]) Unkl[-R,R] ---+ vI [-R,R] for all R~O. The fact that the trajectories (U nk )kEN are solutions of the ordinary differential equation U = - f (u) associated to the smooth gradient field implies the C k _ convergence

V'

IV' f(un(t)) I = 0 we can find a sequence

Unkl[-R,R]

Ck([-R,R]) ---+

E

Lemma 2.39 Let (Un)nEN

J 8

with

In(s) =

lun(r)1 dr

0

Un (r )I

~ E,

by means of (2.45) for t n ~ r ~ to gives rise to the estimate ~

(1) -E

J~;

in (to)

(r)dr

tn

~

~~

J

d(f:sun ) (r)dr = f(un(tn))

V E

COO(JR, M) ,

Proof The first step of the proof is to observe that it is sufficient to show that the elements Un of the sequence converge uniformly toward x and y, respec­ tively, for ItI ----* 00. If this is true, Lemma 2.10 and the associated marginal remark on the dependence to = to(s) imply the existence of a fixed T ~ 0, such that we observe uniform exponential asymptotic convergence with respect to local coordinates at x and y: (2.46)

IU n ,loC x,lI(t)1 ~ cX ,ye->'x,lIl t l

for

ItI ~ T

and for all n EN.

Without loss of generality we can assume that the entire sequence lies within a coordinate neighbourhood of v in p~;~, (Un)nEN c Ol,2(v). Then referring to the local coordinates of the Hilbert manifold

d -(f 0 un)(r) ds

tn

$

such that v E Mt,y is a trajectory in the same trajectory space. Then the sequence (Un)nEN is also Hl,2-convergent,

Hence, we obtain the upper bound from to

M~,y be a weakly convergent sequence

p',2

d(foun)(r) = -1V'f o u n (r)1 2 ds

din -(r) ds

o

Un ~ vEMfx,y

the only step remaining in order to achieve pointwise convergence is to find an upper bound for (in (to) ) nEW Calculating

IV' f

c

Un

tn

din (r) = ds

., WIth v=-V'fov

[-R,R]

for all s E [tn, to] .

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

d(x,un(to)) < ro+ln(to)

V

iteratively for all kEN, thus concluding the proof.

lV'f(un(s)) \ ~

= E and

2.4. COMPACTNESS

Un = expv ~n, ~n E Hi,2(v*TM) ,

~ f(un(to))

to

f(x) - f(y) E

Since (un (t) ) nEN is a relatively compact set for each fixed instant t E JR, the theorem of Arzela-Ascoli provides a subsequence (u nk hEN converging on

the weak convergence

'coo

~n~O

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

2.4. COMPACTNESS

CHAPTER 2. THE TRAJECTORY SPACES

60

Given any neighbourhood U(y), there is an associated to E lR. such that the relation un(t) E U(y) holds for all t ~ to and n E N.

61

in contradiction to the assumption of tk

and un. ~

V

E

Mty.

0

Proof of proposition 2.35:At this stage we finally explore the obstruction to the strong Hl,2-convergence of the given sequence (Un)nEJII of unparametrized tra­ jectories. As we remarked above, Lemma 2.38 leads us to the weak convergence

Let B(y) be an isolating, open neighbourhood of y, that is B(y)nCrit f = {y}. Then the (P-S) condition on f guarantees that

N€ = {p E B(y) I If(p) - f(y)1 < E, lV'f(p)1 < E}, E > 0

Coo (Tn> M) un. Cl~c ----+ vErn.,

(2.48)

and thus to a trajectory associated to the same negative gradient flow,

forms a fundamentaLsystem of neighbourhoods of y. Moreover, we notice that

Uun(lR.)

U =

coo

--+ 00

(:t +V'f) OV=O,

nEJII

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

is a bounded set. This is clear from the decomposition

U=

U{Un (t) II V' f

0

Un (t ) I ~

E}

U{Un (t) II V' f

U

nEJII

0

Un (t ) I < E }

nEJII

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

U{un(t) IIV' f 0 un(t)1 < E} C K:'Y

II V' f 0Un (t) I ~ E}

are

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

Consequently, the set of possible manifolds of trajectories M~,ii' which might contain the element v, is already confined by the condition

nEJII

In consequence of the boundedness of U, the vector field V' f is globally Lip­ schitz continuous on U. This fact together with the equicontinuity of (Un)nEJII from Lemma 2.38 gives rise to the existence of a constant c such that the estimate 11V'f oun(t)I-IV'f OUn(S)11 < c· Jjt"""=Sf

0

un. (s) I >

~

Itk - sl < ­4c2

for

='.

8

~

f(f;)

f(x)

~

f(x) .

Of course, we have to verify that v is an element of such a trajectory space condition as follows: First, relation (2.48) implies the confinement by

f(v(T)) E [J(y), f(x)]

(2.50)

Let us now prove the uniform convergence asserted above by contradiction: If we assume that there is an E > 0 together with sequences nk --+ 00 and tk --+ 00 satisfying lV'foUn.(tk)1 > E for all kEN, then we first deduce, due to (2.47),

IV' f

~

M~, ,y' anyway. We deduce this from (2.48) and the fundamental Palais-Smale

lV'foun(S)1 > lV'foun(t)l-cJjt"""=Sf for all s,tElR.,nEN

E2

f(y)

(2.49)

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

for all t E lR. .

is relatively compact.

i:

such that the solution property (2.51)

v = -V'f

2

for all T E lR. 0

Iv(s)1 ds = f(v(T)) - f(v(-T))

v gives rise to the estimate

~ f(x) -

f(y)

for all T E lR.

00

,

Thus we obtain the finite £2-norm

and thus, second,

J

Ivl 2 ds < 00 and in consequence of this

-00

'j

the asymptotic behaviour

f(Un.(tk)) -f(Un.(tk+8)) =

j

t.+,5

lV'foun.(s)lds > 8.': 2

t.

E3

(2.52)

8c 2

3 E

t-+±oo

= t-+±oo lim v(t) = 0

Finally, by combining the Palais-Smale condition on f with (2.50) and (2.52) we deduce the relation v E M~,ii with respect to critical points which satisfy the estimate (2.49).

From this we extract the estimate

f(Unk(tk)) - f(y) > 8c2

, lim V'f(v(t))

for all kEN, ,,~l;,

, a'

Now, we have to study separately the different cases for x and y. Considering the first possibility, v E Mt,y, we immediately obtain strong Pi:~­ convergence from Lemma 2.39. All the other cases which lead to the splitting­ up into broken trajectories, that is, without loss of generality,

v

E

M~,yl

with f(y)

2.4.3 The Compactness Result for Homotopy Trajecto­ ries Essentially, the statements on the compactness of the trajectory manifolds for the a-operator may also be transferred to the trajectory spaces of the homotopy morphisms. However, since we restrict our considerations to Morse homology, we shall work out the following proofs for the compactness result under more specialized assumptions. This will simplify, for instance, the step of the uniform L 2 -bound.

< f(y') ,

can now be controlled by means of suitable reparametrizations. Let us choose any regular level a en f satisfying f(y) < a < f(y') and a reparametrization of (u nk ), Uk = U nk • Tk = U nk (. + Tk), (Tk hEN C IR ,

Definition 2.40 Let us remember that the coercivity property of the Morse functions appeared as an essential part within the proof of the compactness result for unparametrized trajectories. Turning now to the time-dependent situation of a homotopy between Morse functions we have to guarantee an analogous kind of coercivity. We call a smooth homotopy h = h o. {3 : IR x M ----t IR between the two Morse functions rand f{3 a Morse homotopy, if htO = h(t,') satisfies the following condition:

such that the identity f((u nk • Tk)(O)) = a holds for all kEN. If we ap­ ply Lemma 2.38 once again to this sequence Uk, a further weakly convergent subsequence

_

Ukl

c~c_ ------;

V

is extracted such that it complies with the estimate

(2.53)

f(y) ~ f(V(+oo)) ~ f(v(-oo)) ~ f(y')

= 00,

if

x M, d(xo, X n )

----t

h,

a

.

.

hsO

00 8

=

ah

at (s,·) = 0

for all

lsi

~ R .

Thus, the (IL(X o:) - IL(X{3))-dimensional manifold of the so-called homotopy trajectories (h-trajectory) comprises the solution curves U a {3 of the ordinary non-autonomous differential equation

that this iterative process of reparametrization stops at broken trajectories of order IL(X) - lL(y) at the latest. This can be seen immediately from the inequality

(2.54)

i'(t) =

Vh t

VI + Ih l 1Vhl 2

t

lL(v(j+!) (-t.oo)) < ... < IL(X) , with the ends fixed at the critical points

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

C IR

(tn, Xn)nEN

Now let us conSIder a smooth, regular Morse homotopy f o. :::: fl-'. Apart from the demanded coercivity property it particularly holds that

IL(V(j)(-oo)) -1L(V(j)(+oo)) > 0 ,

=

lim h tn (x n )

n----;.CX)

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

lL(y) < ... < IL(V(j) (-00))

63

2.4. COMPACTNESS

CHAPTER 2. THE TRAJECTORY SPACES

62

Xa

o,(t)

and x{3 of f o. and f{3, respectively.

As to this manifold we are again provided with a compactness result similar to that in the time-independent case. Once again the only obstruction against the strong Hl,2-compactness consists of the CI~c-convergence toward broken trajectories. This time we additionally must distinguish between the IR-invariant trajectories with respect to fa and f/3 and the h o. /3-trajectories. The convergence behaviour of homotopy trajectories is expressed by

.j~

.

~

8Note that this definition relies crucially on the completeness of the Riemannian manifold.

--------

~

.._-- --;;--

1JI~'

"r.n:rm

¥i.

-~~~wro~~~~,,¥;E~~~~~g'1f1'~~~~V-;¥~~~~"it»~i"'illi1i!Alf~1i!f;-f<~~

CHAPTER 2. THE TRAJECTORY SPACES

64

Proposition 2.41 Let (Un)nEN C M

h 13 :

0 XO:'Xfj

2.4. COMPACTNESS

be a sequence of h Otf3 -trajectories.

The computation

Provided that there is no Hl,2-convergent subsequence, there exist critical points Xa = x~, ... , x~

E

Crit(Ja), x~, ... , x1

= xf3 E Crit(Jf3)

65

d dsh(s,"((s))(t)

,

1 ~ k + l ~ J..l(x a ) - J..l(xf3)

h(t,"((t)) (2~4)

(2.56)

and (h af3 -)trajectories j

Jf3

va EM xa,xo: i i+l, Vf3 EM Xf3'X/3 j HI' Vaf3 E

lim h(t,"((t)) = t-->t+

(T~,n)nEN, (TJ,n)nEN

C

with 0

~

i

~

"((t)

c~o

---->

k - 1 and 0

i

j

va<' Un' Tf3,n ~

j

~

c~c

--t

j

vf3

and Un

00 .

Applying the identity (2.54) once again we deduce

IR ,

lim 1'(t) = 0 t-->t+

such that up to the choice of a respectively suitable subsequence weak conver­ gence of the form Un . Ta,n

0

together with (2.55) yields

h af3

Mxa,x13

together with reparametrization sequences

i

2) (ht - VI +l\lhtl 2 Iht 1\lht l2 J

r

i

+ \lh t '1'(t)

C{~)c --t

o

and hence a contradiction to the supposed maximality of "(.

v a f3

Auxiliary Proposition 2.43 There is a uniform L2-estimate for the homo­ topy trajectories with fixed endpoints. It is given as

l - 1 holds. Moreover, the inequality

J 00

J..l(X~) < ... < J..l(X~) ~ J..l(x~) < .. , < J..l(X~) is satisfied.

2

11'1 dt < C(X a , X(3),

E

M

h af3

Xa ,xf3

-00

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

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

00

J

00

2

11'1 dt

(2~4)

lim h(t,"((t))

t-->t+

= 00

,

if t+ is finite. Thus, provided that t+ is finite, it follows that

(2~6)

t->t+

S

00 .

2

00

-00

VI

l\lhtl dt + Ih t n\lh t l 2

­ h(t,"((t))) dt

-00

'1;"'1:'"

I,:,

R

(2.57)

r(Xa<) - ff3(Xf3)

h(t, "((t))dt

-R

f L;IIi

.~,

'·1

lim dd h(s,"((s))(t) =

J

00

Auxiliary Proposition 2.42 Each solution "(: (C(to, xo), t+(to, xo)) ~ M of the non-autonomous differential equation (2.54) satisfies t+(to,xo) = 00, that is the 2-pammeter flow is global in forward time direction.

Proof. Let a maximal solution of (2.54) "( : (t-, t+) ~ M be given. Due to the completeness of the metric space M and the requirement of coercivity (2.53) to the Morse homotopy h af3 it holds that

2

l\lhtl dt ~ 2 1 + Ih t \2J\lht I --.;:

J -J (d~h(T,"((T))(t) +J

-00

-00

(2.55)

for all "(

I:,

'I"""',

;~

:i"\

j

Since we deal with coercive Morse functions, the compactness of the set {x E M I r(x) ~ r(x a )} implies the relative compactness of the set {"(( -R) E !v! I"( E M Xa ,xl3}' As a consequence the continuous 2-parameter flow from Auxiliary Proposition 2.42 yields the fact that

U"( ([- R, RJ) ,EM xa ,xl3

is relatively compact

·",.~c""";-~,,,-,,--

(2.58)

"':;i;~.;:.,;;.;;._-'C . ",.",-"-",,,,,.,..~.

~_,._'~-,

C-' -,;,',...,,"'_, ',-,0"_.":.,...., cc __ ,

__ ~;,~,~:"",,,<.m __ ~ __ •. ".~V"_ ... _ ~~

67

Obviously the divergence Tk - t -00 follows, that is, without loss of generality, < - R. Since rcparametrization leaves the uniform estimate from Auxiliary Proposition 2.43 invariant and since we can further asjume pointwise conver­ gence of subsequences, we are able to deduce the weak convergence

Therefore the uniform bound

K

"';;::;.;

2.4. COMPACTNESS

CHAPTER 2. THE TRAJECTORY SPACES

66

/~~ ...::;:..:=.;:.~.::.-

Tk

K(x a , xf3)

su p { /h(t,)'(t))/I

tE[-R,R],)'EM~:~xfJ}
exists. Finally, the uniform L 2-bound is obtained from (2.57) as c(~a,Xf3) = r(x a ) - ff3(xf3)

+ 2KR

u nk

(2.62)

.

COO •

Tk

~ v' EMf" y,y'

toward an fa-trajectory up to the choice of a suitable subsequence. An analo­ gous reparametrization algorithm thus leads us to the assertion. 0

o

Corollary 2.46 Let us state an important conclusion from Auxiliary Proposi­ Corollary 2.44 In particular, we obtain a compactness result for the h-tra­ tion 2.42. Since the set {x E MI!"(x) ~ !"(x a )} is compact, the same is true jectories as point sets from the auxiliary proposition together with the time­ for the set constant Morse junctions at the ends of the Morse homotopy:

U )'(i)

(2.59)

is relatively compact .

{ )'(+00)

'YEM"ct'''fJ

Lemma 2.45 Let (Un)nEN C M~:~xfJ be a sequence of h af3 -trajectories with fixed endpoints. Then there is a weakly convergent subsequence E

Coo (1Tll M) Jl'l.,

'i~":: I'

1J I:,;

'1; "

v E M~~fJx"(3 0: I

with

r(x~) ~ r(x a ), ff3(xf3) ~ ff3(x~) . fJ

The proof of Proposition 2,41 is carried out now in accordance with principally the same scheme as for the time-independent trajectories. The only difference consists of the distinction between the lR-invariant trajectories and the h-trajectories: Proof of proposition 2.41: It is sufficient to consider the case f(x~) < f(x a ). Once again, let a be any fixed regular value from (J(x~), f(x a )) and choose a reparametrization sequence in accordance with the condition

f (u nk (Tk))

=

a.

}

~: ;,

The compactness result for the parameter trajectory manifold M~ct~:fJ associ­ ated to a generic homotopy of Morse homotopies is obtained according to the same scheme as for homotopy trajectories. We recall that this manifold of A­ parametrized trajectories (provided that we deal with a compact discrete set)

forms the basis for the definition of the chain homotopy operator lJtf3 a . Since

the parameter dependence

~ ~'

o

(2.61 )

'Vh t

VI + ... 0)'

2.4.4 The Compactness Result for

A-Parametrized Trajectories

•

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

(2.60)

"y =

due to the auxiliary proposition. Thus, each critical point X a E Crit fa can be connected to only finitely many critical points xf3 E Crit ff3 via h af3 -trajecto­ ries. In fact, this will be the reason that the homotopy morphism f3 a , which will be dealt with in Chapter 4, is indeed well-defined.

Relying on these preparations we are now able to prove the crucial weak con­ vergence also for the homotopy trajectories.

CJ':c V U nk -----t

I r('Y(-R)) ~ r(x a ),

J­

H>,

=

H(A,', '), H: [0,1] x lR x M

-t

lR

:1\1,:

~.. ;~

:,:; .',If' (~:'

,:~.,;~'

~

j "

is smooth and the parameter set compact, there are analogous uniform esti­

mates. Once again, the only obstruction to the Hl,2-compactness occurs as the

C~c-convergence toward broken trjaectories. Here, these broken trajectories

consist of one H~f3 -trajectory and several fa- and ff3 -trajectories depending

on the dimension and the connected component of the trajectory manifold.

In particular, we deduce the conclusion that M{fctfJ ct, y fJ represents a finite set {(Al,Ul), ... ,(AII,UII )} with Ai i= 0,1, provided that we deal with

the relative Morse index /-l(x a ) = /-l(Yf3) - 1.

CHAPTER 2. THE TRAJECTORY SPACES

68

2.5

Gluing

2.5. GLUING

69

Theorem 3 Given a compact set of simply broken trajectories K

c M{y

x

Mt,z' there is a lower bound PK ~ 0 and a smooth map In the last section we saw how the broken trajectories associated to the time­ independent and time-dependent gradient flow form an obstruction to the com­ pactness of the trajectory spaces. By introducing the gluing operation we now intend to complete the complementary concept of 'gluing' and 'compactness' as was described in the introduction. The gluing operation #P will map simply broken trajectories, that is, pairs of the form

(U, v)

E

Mx,y x My,z ,

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

2.5.1

Gluing for the Time-Independent 'frajectory Spaces

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

#:

K x [PK,oo)

(u,v,p) satisfying: The map #p: K

'----t

t---+

--+

ML

u#pv

Mt,z is an embedding for each gluing pa­

rameter P ~ PK· Moreover, given a compact set R C M{y x unparametrized trajectories, # induces a smooth embedding

'#: R x [PR'oo)

'----t

Mt,z

of

Mx,z ,

such that we obtain weak convergence toward the simply broken trajectory ~

'#' U pV

coo - 10e +

(AU,VA)

as P tends to infinity, P --+ 00. On the other hand we obtain the converse result that any sequence of unparametrized trajectories converging tr: a simply broken trajectory finally lies within the range of such a gluing map #.

The gluing procedure may be sketched as in Figure 2.2.

x

y.d'

~

z·

Figure 2.2: Gluing for unparametrized trajectories

?

(u#Pt V)#P2 W ~ U#Pt(V#P2 W) . The result, which will be sufficient according to our concerns with homology theory, is stated as follows:

In relation to the investigation in the last section this theorem sup­ plies us with a far more detailed characterization of the obstructions to com­ pactness of the unparametrized trajectory spaces Mx,y, at least as concerns the relative Morse index 2. Actually, these obstructions are described topo­ logically as cylindrical ends R x [PR' 00) with the set R of simply broken

70

CHAPTER 2. THE TRAJECTORY SPACES

2.5. GLUING

71

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

Due to the compactness of K we find a finite covering by such neighbourhoods UiL x Uv yielding the constant

The construction of the gluing map is carried out in three steps: First, we establish a smooth pre-gluing map #~, which glues broken trajecto­ ries (u, v) E Mx,y x ;\1y,z rendering a smooth curve in P;:~, such that the image (u#~v )(iR) differs from u(iR) u v(R) by a sufficiently small amount and, moreover, such that the convergence toward the unparametrized trajectories

In the following chapter, on orientation and gluing with respect to Fredholm operators, especially such asymptotically constant curves will playa significant role. These curves allow us to define the simplest kind of gluing operation,l1 namely

U

T(K) = max T(u, v) . {(iL,v)}

ck:c (Au, vA) # 0pv ------->

~(t + p), v(t-p),

t

~0

t~O

We thus obtain

ill/v

is provided for p --t 00. The crucial second step will provide a procedure of associating nearby trajectories from Mx,z to these preparatorily glued curves, which represent approximate solutions of the negative gradient flow, in a unique way for large gluing parameters p. In order to guarantee uniqueness we in prin­ ciple use a normal bundle of Mx,z within P;:;. However, this normal bundle is not supplied canonically. By means of the projection along the noncanonically transversal fibres 9 we find a contracting map. Due to the Banach contraction mapping principle and in consequence the implicit function theorem, we obtain a unique solution of the negative gradient flow if the gluing parameter p is large enough, that is to say that we differ little enough from the original trajectories up to time-shifting. Third and last, we have to verify the embedding property. This is largely owing to the uniqueness lO guaranteed by the Banach fixed point theorem, if p is large enough.

+1

with T = T(u, v) .

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

/3± : :i --t [0,1] , which are defined by

I, /3-(t) = { 0,

t

~-1

t~O

and

/3+(t) =

{O,1,

t

t

~0

~

1

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

#0:

Let us assume as in Theorem 3 that K c M~,y x Mt,z is a compact subset. Then given any pair (u, v) E K, we choose smooth curves (iL, v) E C~y x C:;:z appropriately such that within the Banach manifolds P;:~ and there are coordinate neighbourhoods UiL and Uv containing the previously given trajectories: u E UiL and v E Uv .

Kx

[Po, (0)

(u,v,p)

Pb:;

In particular, let us choose iL and v asymptotically constant at the common endpoint y: iL(t)=y, t~T

v(t) = y, t ~ -T ,for some T = T(u, v)

for all p ~ T

E C'::z

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

The Pre-Gluing Map and the Uncanonical Normal Bundle

9In fact, this can be formulated as a retraction. lOstill referring to the noncanonical normal bundle

(u#piJ)(t) = {

(2.63)

r-+

--t

P;':~

u#~v

by (2.64)

u#~v

=

eXPiL#pv

(/3- . [exp,fil (u)]p + /3+ . [expZ1(v)L p )

for Po = Po(K) > T(K) + 1. Here, (iL, v) are chosen suitably with (u,v) UiL x U" referring to the above covering. The notation .:it, :'llf,J\··

:i: '~r

a

j

Up(t) = u(t + p), t E JR describes the time-shifting that has been already studied above. llNote that this does not concern non-trivial trajectories of the gradient flow.

E

'-,'-. -: : -

~-----~~~~'=" ~-~~ ~--

CHAPTER 2. THE TRAJECTORY SPACES

72

'"

;,

;,."':

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

(2.65) (u#~v)(t) =

up(t), expy[,8- expy1(up) +,8+ expy1(v_p)](t), { v_p(t),

,8-V'2exp~~:~p]

8 8p (u#~v)(t)

up(t), [ { V'2 expy· - ,8+V'2 expy E

v_ p

------ -

= - -------=- - - ­

2.5. GLUING

73

consists of determining the correction term between the actual gluing of trajec­ tories and the already defined approximate pre-gluing in a way which is inde­ pendent of the trivialization. Actually, this amounts to the below construction of a normal bundle LJ...

t ~-1 It I ~ 1 t~1

Throughout the following investigations we will use the notations

x = (u, V, p)

follows from the asymptotic behaviour of u and v and the definition of u#pv. Hence it is only with respect to the lower bound Po(K) that the definition of #0 depends on the choice of the covering of K. Moreover, the two equivalent representations (2.64) and (2.65) yield the smoothness of this pre-gluing map #0. We conclude the differentiability with respect to u and v from (2.64), and formula (2.65) implies the differentiability with respect to P yielding the formula 12

(2.66)

-~~-~=:=::,-~~~

E K x

[Po, 00),

Px,z = P,~::, Po: Px,z ----7 L 2(P;,zTM)

W:)(

= u#~v E

Jf--t"y+V'JoJ,

where we refer to the local representation with respect to w E C'::z:

(2.67) Pw

~

t ~-1 (t), It I 1 t~ 1

=

V'2 exp~l oPo 0 expw : H 1,2(w*TM) cOw

----7

L 2(w*TM)

Since the broken trajectories (u, v) E K are composed of smooth curves, the nonlinear operator Pw ,,- E Fred(w~TM) is well-defined as well. The Rieman­ nian metric (.,.) on M induces well-defined Riemannian metrics

-v_p(t), L 2 ((u#~v)*TM) .

J(~(t),

(~, O~,2

((t) )-y(t)dt

IR

(~,

Remark As we will see in Proposition 2.50 below, already this pre-gluing map #~ has the embedding property if p is large enough, Le. p ~ p(K) for p(K)

for

large enough. This can be seen from the fact that the differential associated to the map (2.64) ker DFu x ker DFv

(~,

()

----7

H1,2((U#~v)*TM)

f--t

V'2 exp (,8- (V'2 exp-1~) p +,8+ (V'2 exp-1 (L p )

0;,2 J

(V't~, V'tO~,2 + (~, O~,2 E

COO

X,Z

C

together with the associated fibre norms (see the appendix)

1 ,2 p X,Z

11·111,2 and 1/.11 0 ,2 on the bundles

H 1,2(p*X,Z TM) c L 2(P*xlz TM) . Thus the smooth map #0; K x [Po, 00) ----7 Px,z supplies us with the Hilbert bundles H = #0* H 1,2(p*xlz TM) and L = #0* L 2(P*x1z TM)

will prove injective for large p. In the following considerations we provide the fundamental struc­ tures which will serve to find the actual gluing map according to the Banach contraction principle. We shall construct suitable Hilbert bundles, a linear glu­ ing version, which is injective, and a normal bundle which is to guarantee the uniqueness of the gluing operation.

on the base space K x [Po, 00). Additionally, the inclusion i : K '---* Mx,y x My,z

induces the Hilbert bundle

It is worth mentioning that if we could cover K by only one sin­

1,2 ( H 1,2)2 = i* (H IMx,y

gle pair of coordinate neighbourhoods, we would be able to reduce the whole problem to the trivialized framework. Then we could renounce this somewhat extensive bundle constructions. The trouble in the non-trivial case, however,

X

H 1,2

IM y ,.

)

on K x [po, 00). According to these bundles Hand L we are able to identify the map Fw ,,- from (2.67) for W x E #o(K x [Po, 00)) via the pre-gluing map #0 : X f--t W x with the fibre restriction of a smooth bundle map

12For the definition of V'2 exp the reader is referred to Definition A.15 in Appendix A.2.

")fJ1•... ;',

"

a ·

,- .------_._--_._-_:-:=--""'­

CHAPTER 2. THE TRAJECTORY SPACES

74

H~O

-

F

\$ /

2.5. GLUING

75

(H 1 ,2)2

L

K x [po, 00)

In this context 0 denotes an open neighbourhood of the zero section, such that the boundary is properly bounded away from the zero section with respect to the norm 11-11 ,2' This is concluded by means of the injectivity neighbourhood 1 associated to the exponential map, the compactness of K and the construction of the pre-gluing map, because the image of the curves u#~ v (i) remains within a compact neighbourhood. This bundle map F: H ~ 0 - L together with the identification Fx = Fwx thus has the explicit representation

=

[(gt + V7 f) oexpupe] (t), V72 expv_)C)-I. [(gt + V7 f) expv_ p C] (t),

V72exPu)O-I. {

0

Fup(e)(t), { FV_p(C)(t), with

eE H

1 ,2

Up'

CE H 1 ,2

V_ p '

t

~-1

t

~

1

t ~ -1

t ~ 1

~r(-00,-11 == ~1(-00,-11 til

"'1[1,00)

-

t

= "'1[1,00)

'

,

and we obtain the identity

Fwx (0) = Fo(w x ) Moreover, the fibre derivative

D 2 F(X)(0) = DFwx (0) : H x - Lx

(~#x()(t)

a

at + X(t) . We shall henceforth use the shorter notation =

~p(t),

[

2

{ V7 exp· (_p(t),

+

exp=~ :~p

J3-V7 2 J3+ V7 2 exp

~1

t ~-1

] (t),

(-p

t

ItI ~1

The fibre derivative V72 exp is taken at the points exp;1 (u#~v (t», exp;1 (up (t»

and exp;1 (v_ p (t», respectively

This corresponds exactly to the linearization of (2.65).

#0

as it is given by equation

The crucial point of the construction of the gluing map can be found in the process which deduces the linearization of the trajectory gluing from this lin­ ear version of the approximate gluing #0. By the linearization of the actual trajectory gluing we mean a projection onto the tangent bundle of Mx,z as a subbundle of HI~x.%' But it is exactly this projection map which cannot be canonical within our construction. We need the concept of orthogonality, which is global with respect to K, namely the normal bundle of Mx,z as has been already described above. So, let us consider the following candidate for a normal bundle:

Definition 2.49 Since D u = DF" and D v = DFv associated to (u, v) Mx,y x My,z are surjective Fredholm opemtors, (ker)2

at 0 is exactly the linearization of F o , which has been already studied as a Fredholm operator of the type

Dx

H

\ffi/

K x [po, 00)

Fwx(~)(t)

L

=

U

E

ker Du x ker D v C (H 1 ,2)2

XEKx [Po,oo)

is a subbundle with constant and finite dimension. Then the following vector space is well-defined for any X E K x [Po, 00):

L1.. = x

{v x E H

1 ,2 x

W

=H I x

(vx,

~#p( )~,2 = 0 } for all (~, () E ker Dv. x ker D v

•

DFwx(O), X E K x [Po, 00)

Definition 2.48 We are able to define the linear version of the pre-gluing opemtion as a bundle homomorphism by

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

CHAPTER 2. THE TRAJECTORY SPACES

76

"{f

Proposition 2.50 There is a lower parameter bound PI ~ Po, such that for any gluing parameter P ~ PI and broken trajectory (u, v) E K the Fredholm operator D x : H x ~ Lx is onto and the composition of #x with the L 2 _ orthogonal projection onto ker D x induces an isomorphism

cPx

= Proj~e~ D" 0 #x : ker D u x ker D v

2.5. GLUING

Therefore it is sufficient to show that, given any broken trajectory (u, v) E K, we can find a lower parameter bound p( u, v), such that cPx is onto for all P ~ p(u,v). This is due to the assumed compactness of K which implies a uniform constant p(K), such that cPx is onto for all X E K x [p(K), 00). Hence we can conclude

-=-. ker D x

dim ker D x

We deduce from this first result that ker D =

77

(c:!:)

U ker D x

~

+ dim ker D v

dim ker D u

J.L(x) - J.L(z) = indD x ~ dimker D x

which gives rise immediately to the identity

xEKx[Pl,oo)

dim ker D x = dim ker D u

is a subbundle of H on K x [PI, 00) with the finite dimension J.L(x) - J.L(z) and that #1(ker)2 : (ker)2 ~ H

and which consequently implies the asserted isomorphism property of all X E K x [p( K), 00).

represents a bundle homomorphism. Hence, we obtain

L.L =

U

+ dim ker D v cPx

for

Therefore, let us choose an arbitrary, fixed broken trajectory (u, v) E

K. Then it suffices to find a constant C < 0 and a lower parameter bound p,

L~

such that the uniform estimate

xEK X [PI ,00)

L.L

ffitop

IIDx~ll£2 ~ c 11~11£2

(2.69)

as a finite-codimensional vector bundle on K x [PI, 00) satisfying the relation

"

holds for any p ~ jj and ~ E

R (#I(ker)2) = H .

L.L

ffitop

This is clear, because there is a

~ E ker D x with (~, R(#x)

In fact, as we shall see below, we are led to the decomposition (2.68)

Lt.

J(

)L2 = 0 , J(

if cPx should fail to be surjective. But owing to (2.69) this ~ E the zero vector.

ker D = H .

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

Let us now prove the estimate (2.69) indirectly. quences (Pn)nEN and (~n)nEN satisfying the conditions

Proof of proposition 2.50: Concerning the central feature, the proof of this nonlinear gluing theorem is essentially patterned on the proof for an analogous gluing operation in Floer homology as can be found in [F-H]. As we already noticed above, the operator D x = DFw " referring to

(2.70)

Pn

~ 00, ~n

E

Lt

has to be

We assume se­

L~n' II~nIIL~n = 1 and IIDxn~nll ~ 0 .

First, we wish to deduce the convergence H

w x = u#ov E pl,2 P x,z

1 ,2

~n ~ 0

is a Fredholm operator satisfying the identity

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

indD x = J.L(x) - J.L(z) According to the assumption that the Morse-Smale condition be fulfilled, the Fredholm operators D u and D v are onto. Hence we obtain the expressions

U (u#~v)*TM

dimker D u = indDu = J.L(x) - J.L(Y), dimkerDv=indDv=J.L(Y)-J.L(z) .

pE[po,oo)

t

';\:,.

I /

~:,,-

This is immediate from the trivial spectral flow associated to the constant curve A y . Now let f3: IR --+ [0, 1J be a smooth cut-off function of the type

as a trivial vector bundle on ffi: x [po, 00 ). Thus starting from a trivialization lP : TMjN(Y) --+ N(y) xIR n we find further trivializations lP u and lPv of u*TM y and v*T M respectively together with

U (u#~v)*TM

lP:

---+

[po,

x ffi: x lR

00)

1,

(3(t) = { 0,

n

pE[po,oo)

and let r n notation

satisfying the relations lPl [1,00) == -
Cl

I

~ sup IlPu#ov(t) ~ tER

C2

lP X : H~,2

--=-. H

,2(1R,lR

with

)

0<

(;1

~ IllPxll op ~

1

~~( ~) rn

rn

.

~

r . C+ IIDXn~nIIL2 + II(D y - DxJ~nIIL2([_rn,rn]) n

~

~ • C + IIDxn~nIIL2 +

+

IIDy~nll£2([_rn,rnl)'

C

=

[~BJ I~I

1

r n

sup IIA Xn (t) - AyIIEnd(Rn)

tE[-rn,r n ]

--+

L 2 (IR "lRn )

sup IIA Xn - AyIIEnd(Rn)

~

JI~nll£2

sup JIA Xn (t) - Ayll

tE[-I,ll

[-rn,r n ]

+ (~n)nEN C H ,2(1R,IRn), II~nllL2 = 1 with l

Here, A Xn is a curve

=

Due to assumption (2.71) and the estimate

n =!!.. H ,2(1R at + A Xn . , lR )

that is

IIDXn~nllL2

--+

sup IIAu(t + Pn) - Ayll

tE[-rn,-l]

0 .

+

sup IIAv(t - Pn) - Ayll ,

tE[l,rnl

A Xn

n

E Coo (ffi:, End(IR ))

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

determined by the trivialization lP Xn ' Referring to the trivialization lP y TyM --+ lR n we obtain

A y = lP y ' D(V'2 exp;1 .(V' f

0

lim IIDy(Wn. ~n)IIL2

n--->oo

expy)) . lP;1

Dy

a

at + A y

=

induces an isomorphism

D y : H 1,2

131'f>pl is the norm of 'f>p : TpM upon TM.

-=-. lR

n

=0 .

Therefore, the isomorphism D y from (2.72) yields the Hl~;-convergencewe wish to verify. Finally, this result can be transferred back to the non-trivial case, that is

as a constant operator which is conjugated to the Hessian H 2 f(y), so that the Fredholm operator

(2.72)

dt·

r~ II~(,·: ')~nll£2

(;2 .

Next, these uniform estimates enable us to consider assumptions (2.70) above without loss of generality for the trivial case

(2.71)

~(f3rn )(t)

f3r n(t) = f3( ~), s.t. rn

~

Concerning the class L 2 we are led to an analogous result.

D Xn

be the sequence r n = ~Pn' Additionally, we choose the short

IIDy(Wn . ~n) II £2 11(f3~n) . ~n + f3r n . Dy~nll£2

for all P ~ Po ,

P

n

It I ~ ~ ItI ~ 1

These preparations give rise to the following series of estimates:

and for all p ~ Po the isomorphisms l

--+ 00

== lPvl (-00,-1]

such that we obtain the uniform estimate 13 0<

79

2.5. GLUING

CHAPTER 2. THE TRAJECTORY SPACES

78

(2.73)

1 • 2 ([ c H Xn -rn,rn ] )

<"n

---+

0 .

By this conclusion, that the supports of ~n become more and more concen­ trated at the asymptotic ends as n increases, we are now able to transfer the estimate IIDXn~nll£2 --+ 0 Xn

--=-. L 2 with respect to the Riemannian metric (.,.)

t

CHAPTER 2. THE TRAJECTORY SPACES

80

for the varying Fredholm operators to the fixed operators Dv. and D v:

IIDu(,81-Pn~n'-Pn)IIL~

11;31~nll£2

+ 11,81 Dv. pn ~nIIL~pn

"Pn

11;31~nll L 2xn +

11,81 DXn~nll£2Xn

II~nIIL~J[-2,-1,J) + IIDxn~nIlL~n

- ::;;

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

IIDu(,81-Pn~n,-pJIIL2 --> 0 .

" Since Dv. is a Fredholm operator, this amounts to the existence of an (); E ker Dv. with _

H 1 ,2

-

(};Pn II £2

--> 0 .

V-Po

--> 0,

(); E ker Dv. ,

-->0,

'YEkerDv

'

The contradiction with the former assumption is now caused by trying to piece together all these different conclusions in the series of identities: 1

lim II~nIIL2Xn

= 3

'Y

=0

-

F

\$ /

L

Additionally, we want expw 'Y(X) to comply with the geometrical convergence toward the broken trajectory" (u, v) as the gluing parameter P tends to infinity, P --> 00. Since we are provided with the convergence property for u#~v due to the construction, we shall be satisfied with a sufficiently rapid convergence for the correction term, 'Y(u#~v)-->O as p-->oo.

)

lim (,81~n, ~n) £2 n

Xn

+ lim (,8~l~n, ~n) L2 n

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

Xn

as supp(l -,81 - ,8~1) C [-2,2] (2.74)

-

0

n--+(X)

lim (,81~n + ,8~l~n + (1- ,81- ,8~l)~n, ~n )L2" n n (2:2:

satisfying F

K x [Po, 00)

Summing up, we observe the following approximate behaviour:

11,8~l~n-,8+'Y-pnIIL:n "n

for X E K X [PI, 00)

finally represents a gradient flow trajectory. That is to say that we are searching for a section

H-:JO

"Pn

11,81 ~n - ,8- (};Pn II L2

expw" 'Y(X)

with respect to the formerly defined bundle map

In an analogous manner we are provided with a 'Y E ker D v satisfying the relation 11,8~1 . ~n - 'Y-PnIIL2 --> 0

(2.74)

Up to now we have merely analysed the pre-gluing #0 as an approximate version of the gluing operation. However, the thus defined curves W x = u#~v are not yet trajectories for the negative gradient flow. Resorting to the normal bundle L.L constructed above we now try to find the correction term for #0 with respect to the actual trajectory gluing. Moreover, this correction term has to be provided globally on the entire base space K x [PI, 00). This requires K to be compact. In other words, our aim is to find a 'correction section' 'Y in the bundle H on K x [PI, 00) in a unique way, so that

'Y : K x [PI, 00) --> H

,8l-Pn . ~n,-Pn ~ (); In particular we obtain the convergence 11,81 . ~n

81

The Trajectory Gluing

IIUi-)l-pn~n'-Pnt~ + 11,81-PnDu~n,-pnIIL~

::;;

2.5. GLUING

. (~\ hm}J (};Pn' ~n 1£2 n Xn

(+ + l'1m n ,8 'Y-p

n

,~n

\IL2 Xn

lim ((};#Xn 'Y, ~n)£2Xn n 0, as

~n E

L;n

The proof of the assertion follows.

for all n E N

o

"" D 1£l- : L .L --=-. L2

82

CHAPTER 2. THE TRAJECTORY SPACES

2.5. GLUING

83

(un#~m vn)*TM «Pm,n. (u#~m v)*TM

Thus, considering the inverse map14 G: L 2 ~

L-i-, G = (D 1L-I-)-l

\m/

,

we can extract the following estimate from a refinement of the proof of Propo­ sition 2.50:

iR Lemma 2.51 There is a lower pammeter bound P2 ~ Po and a uniform constant C K ,l > 0, suc.h that the isomorphism G satisfies the estimate

IIGx~lll,2,x ~ C K ,l 11~llo,2,x for all X E K

X

[p2' (0), ~

E

L~ .

Proof. The decisive feature within this proof is the uniformness with respect to the broken trajectories (u, v) E K. An equivalent statement of the assertion via (= G x . ~ E L~ is given by:

1I(1I1,2,x ~ CK,l IID x . (1I o,2,x

for all (E L~, X E K

X

[p2, (0) .

In order to carry out an indirect proof similar to Proposition 2.50 we assume a parameter sequence Pn --+ 00 and associated broken trajectories ((Un, vn))nEN C K together with

relying on the definition 1
II D Xn (n II O,2,Xn

1 •2

-----t

(u,v)

yields the convergence of the induced Banach space isomorphisms

-=. H Xn1,2

1

and L 2Xnn

-=. L 2Xn' respectively ,

with «pn(s)(t) =
and

= 1 and

H

(un,V n )

II
satisfying the conditions lI(n 111,2,Xn

Let us henceforth use the notations Xnn = un#~n Vn and Xn = u#~n v. Then the explicit construction of the pre-gluing map #~n together with the H 1,2. convergence

(n E L~n' Xn = un#~n Vn

(2.75)

0

--+

0 .

II
as n goes to infinity, n

In contrast to Proposition 2.50 the main difficulty lies in the estimate for an inconstant sequence ((Un, vn ,)) nEN instead of a fixed pair (u, v). Therefore we construct an appropriate transformation to the situation of such a fixed pair. Due to the compactness of K, we can assume without loss of generality an H1,2·convergent sequence, that is in particular a CO·convergent sequence

0

D Xnn - D Xn

0


Xnn

-----t

L2 ) -----t l

Xn

1

0

--+ 00.

Moreover, given any f E (0,1) the convergence

=

1 for all (n E L~nn

into the estimate

(un, Vn )

CO -----t

(U, v) E K . 1- f

Considering the contractability of the curves u, v and

u#~v

neighbourhood U of u(iR) U v(iR), a trivialization
Xn,m(iR) = un#;m vn(iR) c U

we find an open ..::.

U x jRn and

holds for all m, n ~ no .

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

~ II (prL~n

0

~ 1+f

for all n ~ no. Piecing everything together we come to conditions similar to those in the proof of Proposition 2.50. Now confined to the fixed pair of trajectories (u, v) and using the notation

~n = (prL~n

0

L~n

we obtain the conditions

111~nliI,xn -

11

~

E,

IIDXn~nll

--+

0, Pn

--+ 00 ,

--.-.------.-.---=.=---=.:-~.="'C• • ' " " S # L = ' .7.~'-'=,"_'"_---::______=~~~--:-'·-----

:-'-:.- : ..: - _ .--- -~.:.-

--"' __.:

'--.

--- ~~--f.i:f';;--':_':; ._- .._ .... _-_..- -•.:-.-:" -=-~-=-=--=-~. ::::::.':::=~~~~::~~-~~:~~~:::'~~;;:~~~~~~::~~~~~~~~L,"'O;O=-~:~~~~

CHAPTER 2. THE TRAJECTORY SPACES

84

as

I (prR(#xn)

0

((n)11

<1>n)

1,Xn

~0

lim II~n Ilo,xn

=0

for some suitable constant C

~

C lim

.

85

'1': E ~ E, 'P(x)

+ C)

11'P(x) II ~

II~nllo,xn

1I'P(x) - 'P(y)11

lim II~nllo,xn = 0

1 - t ~ lim II~nlll'Xn

as stated above. Thus we have proven the uniform estimate of the assertion. D

F be a smooth map between Banach spaces of the

+ D f(O)

.x

+ N(x)

Let us further assume G: F

~

E to be a right inverse,

f(x) = f(O)

D f(O)

0

G

with dim ker D f(O) < 00

=

idp ,

and let C > 0 be a constant such that the contraction property IIGN(x) - GN(y)11 ~ C( Ilxll

+

Ilyll) Ilx - yll

is satisfied for all x, y E B f (0) with t = 5~' Then the initial condition IIGf(O)11 ~ ~ implies the unique existence of a zero point of the map f Xo E Bf(O) n G(F) , which additionally fulfills the estimate Ilxoll ~ 21IGf(0)11

1

1

t(2 +Ct) = t(2

1

+ 5) < t

=

IIGN(x) - GN(y)11

~

C 2t Ilx ­ yll =

~

1 211x-yll

2

5 Ilx ­

yll

Hence we find a unique Xo E Bf(O) satisfying the identity 'P(xo) tionally we obtain the estimate

Let us now introduce the zero point lemma which is central within our Banach space calculus. ~

t

2 + Ct 2 =

due to the assumption. Moreover, 'I' contracts on Bf(O) as we immediately see from

contradicting

Lemma 2.52 Let f : E form

= -G(J(O) + N(x)) ,

we easily calculate the inclusion relation, as x E Bf(O) yields the estimate

.

> O. But this would imply the inequality

lim II~nlll,Xn ~ (1

~-~::.-~.=:-:-=~:..;.::=.. -~:;;;y -:;ii~~~~~~~~~~~~'="'~~~~i:i.<~"i=.~~~~~~~ __

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

On the other hand, the convergence IIDxn ~nllo ,Xn ~ 0 leads to the estimate lim Iltnllo,xn

~:,,'-""'-'_ ----::=-"_ .....'l'_. _ •

2.5. GLUING

Hence we are finally able to conclude as in Proposition 2.50: n->oo

---.• _ ---_ ~

Ilxoll -

= Xo.

Addi­

1 11'1'(0)11 ~ 11'P(xo) - '1'(0)11 ~ 2 Ilxoll

Thus we deduce the inequality Ilxo II < 2 IIG f(O) I and complete the proof.

D

It is by means of this fundamental lemma that we shall later prove the existence of the necessary correction section "( : K x [pk, 00) ~ L1-. Let us choose a fixed X E K X [P2, 00) and let us consider the smooth mapping between the Banach spaces F . H 1 ,2 ~ L 2 x· X X ' that is, explicitly, 15

Fx(x) with Fx(O)

=

'Vtx

+ 8(x). Wx + 'V2 exp(x)-1

. ('Vf

0

expw x) x

= F(w x ), DFx(O) = D x ' Considering the expansion Fx(x) = Fx(O)

+ Dx(O) . x + Nx(O, x)

we compute the difference of the nonlinear terms as

Nx(x) - Nx(Y) =

Fx(x) - Fx(Y) - Dx(O) . (x - y)

= (Dx(Y) - Dx(O)) . (x - y) + Nx(y, x - y) 15For the notations used in this formula the reader is referred to the discussion in Appendix A.2.

2.5. GLUIUG

CHAPTER 2. THE TRAJECTORY SPACES

86

Thus it is sufficient to estimate

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

U

(2.76)

IIF(wx)ll p =

wxOR") relatively compact,

Ilxlkx + Ilylkx) Ilx - ylll,x

II~pIIHl,2([-1,O]) + 11(-pIIHl,2([O,1]))

Is(t)1 :::;: c(s)e- mt , t ~ 0 ,

The constant C K ,2 is a function of the C 2 - norm of \7 2 exp on the compact clo­ sure of the set in (2.76). The required estimate thus follows from this inequality together with Lemma 2.51 and with CK = CK,l . CK,2,

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

Ilxl11,x + IIYlkx) Ilx - ylkx = IIGxF(wx)111,x :::;: CK,l IIF(wx)llo,x '

IIGxNx(x) - GxNx(Y)111,x :::;: CK ( (2.78) and IIGxFx(O)lkx

const (

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

yields an estimate which is uniform with respect to X,

IINx(x) - Nx(y)llo,x :::;: CK,2 (

IIF(wx)llp([-l,l]):::;: const IlwxI H1 .2([_1,1])

:::;:

XEKx[po,ao)

(2.77)

87

1I<,IIL'(r-',o])

for all X E K X [p2, 00) .

~

U

IW + p)I'

dt)! "co",l(o, 0) e-

Fx : H x

estimate

o

----+

Lx

and

G x : Lx

holds for all P ~ Po and (u, v) E K. ,(X) E B,(O)

= u#~v for the

Fx(,(X)) = 0 and

n L~

lIi(x)lkx < 2CK,1

IIF(wx)llo,x '

and thus

that is, without loss of generality,

u(t) = expy ~(t), v(t) = expy ((t),

t

~

:1.

-p-1

t:::;: p+ 1

,

lIi(x)lkx < c(K) e- mp

= ~ + X: H 1,2(1R, IRn ) ----+ L(lR, lRn ) X : IRn ----+ IRn , DX(O) = Hessian of F at y

it

F

';,:.~' . ~~;

'r~

(-p .

',;h

:I.·.:'tr.··.·.. ..

.

Therefore it remains only to verify the smoothness of this correction section ,. Lemma 2.54 The unique section, : K F 0 , = 0 is smooth.

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

+ (3+.

H x , R(G x ) = L~

complying with

F(wx)(t) = 0 for It I > 1 ,

= (r· ~p

----+

satisfy the assumptions for the fundamental Lemma 2.52 and guarantee the unique existence of a correction term

mp IIF(wx)ll o,x : :;: ae­

Wx

elc. ,

Summing up, we can state that there is a lower parameter bound PK, such that, given any X E K X [PK, 00), the mappings

Lemma 2.53 There are uniform constants m> 0 and a> 0, such that the

Wx

"

yields the completion of the proof.

Finally, it remains to show that the deviation of the approximately glued tra­ jectories from the actual gradient flow trajectories tends to zero as the gluing parameter P goes to infinity. This follows immediately from the next lemma.

Proof. Due to the construction of the pre-gluing operation broken trajectories we immediately obtain the identity

m

Proof. we are that is diately

X

[PK,OO)

----+

L1-

n B,(O)

solving

Since we have already solved the existence and uniqueness problem, now able to regard the same zero point problem in local coordinates, with respect to a local trivialization of L1- and L 2. Then we imme­ obtain the assertion from the implicit function theorem together with

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

IID'Y(x)11

~

lI'Y(x) I

const(K)·

Kc

for all X E K X [PK, 00) .

we define the induced gluing operation on this set by

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

#:

K

X

[PK,oo)

X

(2.79)

->

Mx,z

t----+

expl.l#OV 'Y(X) p

Mx,y x My,z compact,

u#pv = [u#pv] = (U#pV)T(U,V,p) , where T( U, v, p) is determined uniquely via the condition

f( (U#pV)(T(U, v, p))) = a

This means that, after the gluing of the parametrized trajectories (uo, vo) E Mx,y x My,z, we have to execute a gauge-shifting or reparametrization by d ...), in order to get an unparametrized trajectory u#pv E Mx,z regarding

The next step is to derive the embedding property for fixed parameter P ~ PK from the embedding property of the pre-gluing map #~ and the exponential convergence IID'Y(x)II, lI'Y(x) II ~ conste- mp •

the above identification.

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

Now, we are able to prove the embedding result in version of

Proposition 2.56 There is a lower parameter bound Pj(, such that the un­ parametrized gluing map

(u, v) E Mx,y x My,z

#: K x (Pj(,oo)

The Embedding Result and the Convergence

(u,v,p) In order to arrive at the statement of Theorem 3 we are finally going to prove the embedding property and compatibility with complementary weak convergence toward the original broken trajectory as regards the spaces of unparametrized trajectories. At first, we define th~luing operation which is induced on the unparametrized trajectory spaces Mx,y.

Mx,y x lR (u, t)

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

D#(u,v,p)

-..::.. Mx,y t----+

"

Uo

M~,y, "

==

u

with respect to a ft:xed regular value a from (J (y), f (x) ). By reverse argumen­ tation we can also consider the projection map

[. ] : Mx,y

->

(u,v,p) ,

(a): Due to the compactness of K it is sufficient to find a lower parameter bound p(u, v) for each broken trajectory (u, v) E K in such a way that the regularity of D#(u, v, p) holds for all P > p(u, v). Since the regularity of the differential of a continuously differentiable map defined upon a finite-dimensional manifold is a so-called open condition,16 this lower bound can be found as an upper­ __ )E K~ p(u, v) exists. semicontinuous 17 function on K, so that p(K) = max( U,v

~

Mx,y

is an isomorphism for each triple

and (b) the injectivity of the gluing map.

Ut ,

for which we implicitly use the identification (2.80)

Mx,z t----+u#pv ->

is a smooth embedding.

Definition 2.55 Let us consider the diffeomorphism already analysed above,
.

16In other words, each regular point (u,v,p) E K x [0,00) with regard to the differential has a neighbourhood in K x [0, 00) consisting of regular points. 17p upper-semicontinuous means: X n ---+ X => limp(x n ) ~ p(x).

D#

Mx,y , ;~'

11

. '1. . .1

"

:'

2.5. GLUING

CHAPTER 2. THE TRAJECTORY SPACES

90

for all n E N. Due to the construction of the gluing operation in the last section we compute

Hence, we shall prove the existence of such a bound for any arbitrarily chosen fixed pair ('11, v). Due to the identity dim(Kx(PR'oo))

=

=

91

J.L(x)-J.L(y)-1+J.L(Y)-J.L(z)-1+1 J.L(x) - J.L(z) - 1

D#Pn(u,V)' (~,()

V I exp (Xn, 1'(Xn)) . (~#Pn ()

~

+ V2exp( ...) . (D1'(Xn) . (~, ())

dimMx,z so that the sequence (1ID#Pn Il op )nEN C IR turns out to be bounded. Since the sequence ft(U#PnV)T(Pn) is also bounded uniformly as we see from the equation

it is sufficient to verify the injectivity of D#(u,v,p) for all P> p(u,v). This will be accomplished indirectly. Let us henceforth use the shorter notation '11 = U referring to the above identification = Ma. From the analysis of the unparametrized trajectory spaces we obtain the decomposition of the tangent space

M

a

~

(2.81)

Tu.#pvMx,z

at

A#~

U

=

Ilft(U#pn V)T(pn)t2

Tu.#pvMx,z E91R at (u#p v )

AE M~X,z (2.80) Ma = x,z

pV

C

M

x,z

a

f(y). Then we compute the

ProjT M .. ,

X,%

(D#p(u,v).(~,()+D#p(u,v).(u,-v).t)

Thus, assuming that D#(u, v, p) is not injective, we find non-vanishing tangent vectors (~, ( , t) :j:. 0 satisfying

D#p(u, v) . (~, ()

+ D#(u, v) . (u, -v) . t E IR· :t (u#p v )

V I exp (Xn, 1'(Xn)) . Xn

+ 11(11 2 + t 2 =

(2.84)

VI exp (Xn,1'(Xn)) . (~n#Pn(n

+ u#Pn(-v) + tn:t(u#~n v)) + V2 exp(...) . (D1'(Xn) . (~n + U, (n - v) + t nVn(Xn)) = 0

Exploiting the well-known identityI8 (V 2 exp-I oV I exp)(p, 0) = idTpM and the exponential decrease

1 .

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

--+

00, ((~n,(n))nEN bounded and (tn)nEN

C

1R

satisfying the identity (2.83)

D#Pn'

(~n,(.n) + D#Pn' (u,-v) =

+ V2exp (Xn, 1'(Xn)) . V n(Xn)

yields the identity (2.83) as

Without loss of generality we can carry out the normalization

11~112

a

at(u#Pn v ) = at (exPxn 1'(Xn)) =

D#( '11, v, p)(~, (, t) =

Jf(x) - f(z) ,

and

#-

(2.82)

=

D#Pn . (~n,(n) + D#pJu, -v) VI exp (Xn, 1'(Xn)) . (~n#Pn (n + u#Pn (-v)) + [V2exp (Xn, 1'(Xn)) 0 D1'(Xn)] . ( ... )

w

:t (u#p v ) = :t (U#pV)T(U.,V,p), f(u#pv(r)) = a . =

lI%t(u#Pn v )ll£2

we deduce the same for the sequence (tn)nEN. A detailed calculation involving the formulas

This follows from the construction of the induced unparametrized gluing oper­ ation with

Without loss of generality we can assume a differential of by means of (2.81) as

=

~

Ih(Xn)/I, IID1'(Xn)11

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

(2.85)

tn' :t(U#pn V)

~0 ,

Ilxn#PnYn

18See Appendix A.2.

+ u#Pn (-v) + tnft(u#~n v)III,2

--+

0 .

CHAPTER 2. THE TRAJECTORY SPACES

92

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

Ilxn + u(1 + t n ) II H,,2« -OO,Pn -1]) n--+oo. lim llYn + V (t n - 1)IIH,,2([I-Pn,OO)) n--+oo

2.5. GLUING

93

together with the exponential decrease 'Y ( Xi,n )

lim

0

_

-

X

n + (1

+ tn) . U

H

'

,2

0 and Yn

-----+

II(Ul#~nVl)rl,n -(U2#~nV2)r2,,,lll,2 ~O .

(2.87) H

1) . v

+ (tn -

,2

' -----+

0

Taking into account that, due to the assumption and the decomposition (2.81),

the relations X

n E TuMx,y, Tu,Mx,y

Yn E TvMy,z, Tv,My,z

-

-

n!R.u = {O}, n!R.v = {O}

0 and

tn

-+

According to the construction of the pre-gluing map #0, the fixing of the time-parametrization f(ul(O)) = f(u2(0)) gives rise to the asymptotic synchronization

'Tl,n - 'T2,n

-1

'11 1

-+

0

and

tn

-+

+1

(Ul,n, Vl,n) = (U2,n, V2,n) ,

holds for all n

respectively. Hence, we deduce a contradiction.

(b): We now have to verify the injectivity of

if- : R x [Pk, 00) -+ Mx,z'

Let us

now assume the existence of sequences

Pn

-+ 00

and

(Ul,n,Vl,n) -:j:. (U2,n,V2,n) E

~

no in contradiction to the assumption.

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

Proposition 2.57 Given a broken trajectory (u,v) from Mx,y x My,z, each arbitrary increasing sequence of parameters Pn -+ 00 together with the gluing op­ eration for unparametrized trajectories induces the geometrical convergence 19

if-

for all n EN,

A

[Ui,n#Pn Vi,n] = (ui,n#p"Di,n)r',n ' i = 1,2 with f ((.. ')ri,,, (0)) = a. Resorting again to the compactness of consider without loss of generality the convergent sequences -+

Ui, Vi,n

-+

Vi; i = 1,2 .

Then the representation u·'Z.ln # Pn v·1.,n A

A

--

Cl:c

A

A

R

we can

The converse is also true: Any sequence of unparametrized trajectories converg­ ing to~a simply broken trajectory finally lies within the range of such a gluing map #. Proof. The C~c-convergence toward '11 and V arises from a respectively ap­ propriate choice of the value a with regard to the identification Mx,z = M~,z' Firstly, let us choose a value satisfying f(z)

-- exPXi,n 1

A

U#p"v -----+ (u,v) .

where we refer to Definition 2.55, that is,

Ui,n

0

R,

which satisfy the identity

[Ul,n#Pn Vl,n] = [U2,n#Pn V2,n]

= '11 2 = '11 and VI = V2 = V

However, knowing from part (a) that if- is a local diffeomorphism, in particular at ('11, v) E R, we find an initial index no EN, such that the identity

and

Yn

0

-+

and thus to the identities

hold and that TuMx,y and TvMy,z are finite-dimensional, and thus locally compact, we observe that the zero-convergence in (2.86) leads to X n -+

0

-----+

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

,

that is (2.86)

n--+oo

. , Xt,n . -- u·'Z-,n #0Pn v·'Z.,n X'l.,n

( )

A

A

< a < f(y) ,

19i.e.: weak convergence with respect to a suitable sequence r n of reparametrization

CHAPTER 2. THE TRAJECTORY SPACES

94

that is

2.5. GLUING

95

Conclusions a

((U#PnV)(O))nEN C M . Due to the compactness of M a we can find without loss of generality a limit point Xo E M a , ,~ ' ( ) n--->oo u#Pn v 0 -----> Xo Considering the unique trajectory s E COO(JR, M) satisfying

s = -Vf ° sand

s(O)

= Xo

,

we are led to the weak convergence

'# Pn V'"'

U

C{~c

----->

s

At the end of this section, regarding the application within our homology theory in question, we wish to discuss some consequences of the compactness-gluing­ complementarity which has been proven now. Actually, we shall consider the unparametrized trajectory spaces of second order and the simply broken tra­ jectories. This means that we restrict ourselves to the relative Morse index f..L(x) - f..L(z) = 2. Then we are able to classify the conn~ed components M~,z of the one-dimensional manifold without boundary Mx,z by the diffeomor­ phism type of 8 1 and (-1,1), respectively. We obtain the following Corollary 2.58 Let ¢ : M~ z ~ (-1,1) be a diffeomorphism. Then there are exactly two different broken trajectories

where the reparametrization sequence Tn for

u#.Pn v = (u#Pn vk = exp(fJ.#oPn i1)T n [r(Xn)]Tn

(Ul,Vl)

f= (U2,V2), (Ui,Vi)

E MX,Yi x Myi,z; i = 1,2

is determined by

f((U#PnV)(Tn))

=

satisfying

a .

f..L(x) - 1 = f..L(Yl) = f..L(Y2) = f..L(z)

The exponential decrease of ,(Xn) analysed above implies the estimate

b(Xn)(Tn)1

~

const· e- aPn

-t

+1

,

and, given any small E > 0, we jind gluing parameters PI (E), P2 (f) E JR, such that we obtain the identities

0

and thus the convergence (U#~n V)(Tn

) -t

(¢O#.)({Ul} x {VI} x (PI, 00)) = (-1,-1 +f)

Xo .

Hence, due to the construction of the pre-gluing operation, the condition f(xo) < f(y) gives rise to the limit point lim V(Tn - Pn) = Xo ,

n--->oo

that is,

s=

V.

An entirely analogous argument now starting from the assumption f(y) < a < f(x) yields weak convergence to the other part of the broken trajectory, coo U'# Pnv'~' U • The converse statement is an immediate result from the uniqueness provided by the Banach fixed point principle by means of which we developed the gluing maps. If we are given any sequence of trajectories with fixed endpoints which converge weakly toward a specified simply broken trajectory, the elements fi­ nally lie within a H 1,2- neighbourhood in which the above contraction mapping principle (see Lemma 2.52) works. Hence, these elements have to lie within the range of the gluing map belonging to this specified simply broken trajectory. ~

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

and (¢ ° #.) ({ U2} x {V2} X (P2, 00)) = (1 - E, 1) .

Proof. The boundary property is a consequence of the gluing theorem. The fact that the broken trajectories cannot be identical arises from the uniqueness of the correction term ,(Ui#~Vi) with respect to the uniquely supplied bundle £1. with regard to large gluing parameters. This is the same argument which was used for the last statement of Proposition 2.57. 0 From this corollary we gain an equivalence relation for simply broken trajectories with relative Morse index 2. This equivalence relation will prove crucial for the construction of Morse homology. In fact, it is the kernel of what we called in the introduction the cobordism relation for trajectory spaces. Definition ~.59 Let us dejine-ihe set!!i simply broken trajectories with fixed endpoints Mx,z = {(u,v) E Mx,y x My,z I f..L(x) - 1 = JL(Y) = f..L(z) + I}. The1J:J....the gluing cobordism from Corollary 2.58 induces an equivalence relation on Mx,z by

(Ul,Vl)""" (U2,V2) ~ (Ul#P1Vl) ~ (U2#P2V2) in Mx,z

¥¥£¥4¥

!

!!I!!!"'-

: _

CHAPTER 2. THE TRAJECTORY SPACES

96

decompos~

RX

Uh

CI~c # Pn V_ ---+

(Uh,

v)

.

E

h M x",x/3

X

f/3 M X/3,Y/3

t :::; P - 1 (3=p[exp;;(uh)] (Uh#pV)(t) eXPX/3 ( (3+ [ -l()J + -p eXPX/3 v -2p ) (t), It - pi :::; 1 { V-2p(t), t ~ P+1

(2.88)

_

In this framework, the lower bound Po for the gluing parameters is now addi­ tionally bounded below via the time interval during which the Morse homotopy

tiS

r is active, that is Po

~ f{3 .

')

Uh, V

00, we are

Uh(t),

o

'\

r

(

---7

by

Therefore, we must distinguish three different types of gluing operations:

We shall obtain the result for the mixed broken trajectories analogously to the first gluing theorem. Let h t be any regular Morse homotopy

V, p)

Proof· Essentially, the construction of this gluing operation for mixed broken trajectories differs from that in Theorem 3 merely by the definition of the pre­ gluing map. Instead of shifting both parts of the trajectories by the value P in opposite directions regarding the time-parametrization, we now define the pre-gluing map #0 for the mixed broken trajectory

~.~

e gluing of broken trajectories which do not contain any shifting-invariant pieces at all.

h

such that, given any arbitrary sequence of gluing parameters Pn provided with the weak convergence

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

e gluing of broken trajectories, which consist of shifting-invariant pieces as well as of trajectory pieces corresponding to the time-dependent gradient field, called mixed broken trajectories, and

."IllIIII

~Mx",Y/3 1--+ Uh#pV ,

[Pj(,oo)

(Uh,

ueT=U T

e gluing of shifting-invariant trajectories, as analysed above,

n~'

~/3

#:

The above defined equivalence relation for simply broken trajectories of the time-independent gradient flow yields the crucial argument in the proof of the chain complex property f)2 = O. We now require the corresponding gluing results for the time-dependent and the >.-parametrized situation, in order to prove the homotopy invariance of the Morse homology in an analogous way. The essential difference between the earlier gluing operation and the gluing operation which is to be constructed later, is a lack of invariance under time­ shifting

for It I ~ R .

II"IIIIIII!II

97

2.5.2 Gluing of Trajectories of the Time-Dependent Gradient Flow

f)t (V'h t ) = 0

'M1!I!IIIll!11'"

Theorem 4 Given a compact set of mixed simply broken trajectories K C X M~/3'Y/3' there is a lower parameter bound Pj( and a smooth embed­ ding

M~",x/3

the set Mx,z into pairs of different equivalent broken

f)

_!iiI! k_UlR:rrr

UIillJIlIlIIMIIllli!II"IR

2.5. GLUING

This means that these respective broken trajectories correspond to the ends of the same pathwise connected component of .Mx,z, so that they are cobordant in the sense of 'compactness' and 'gluing '. It is obvious from the possible diffeo­ morphism types of one-dimensional manifolds without boundary that it holds that #[('11, v)J = 2 for all ('11, v) E Mx,z . We can thus trajectories.

il"IllI!£IlIl!'NM!IImIW

~

~ f{3

R(h t ).

The construction of a smooth gluing operation for these trajecto­ ries is accomplished by steps entirely analogous to that of the former gluing operation. What proves decisive for transferring the essential stages of the con­ struction, is the invariance under time-shifting of the H 1 ,2_ and L 2 -products along the trajectories. It is worth mentioning, that this time we can consider equivalently the gluing operation as an embedding

#p : K ~ M~"'Y/3 of compact sets K C M~",x/3 X M X/3,Y/3 for fixed gluing parameters P ~ PK as well as the embedding stated above,

#:

Rx

[Pj(, 00)

~ M~"'Y/3 .

CHAPTER 2. THE TRAJECTORY SPACES

98

99

In order to obtain the correction term for an h-trajectory by means of a con­ traction map, we choose a suitable homotopy between fa and f{3, namely

Once again, it is immediate from the representation

Uh #Pn 1; = eXPUh#oPn ii 'Y(Xn) together with 1'Y(Xn)(t) I ~ conste- aPn ~ 0, that the original trajectory (Uh, 1;) may be reproduced from the weakly convergent sequence

( ') Uh # pn v,c;:c ------+ Uh, V .

o The respective result for mixed broken trajectories of the form h

~

2.5. GLUING

(il, Vh) E Mx""y" x M Y",Yf3 is obtained likewise.

(290) .

ha"Y(t) R

= {

a h {3(t + R,·), h{3"Y(t - R,'),

t ~0 t ~0

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

Broken Pure h- Trajectories

Corollary 2.61 Given the situation of isolated h-trajectories, that is For the case of the gluing of broken pure homotopy trajectories, that is a hf3 similar operation which glues trajectories (u a{3, U{3"Y) E M~"f3x and a, ~ X M x/3''Yx maps them into the trajectory space M~"'Yx , we should present a separate co, 'Y approach. The crucial feature in this situation is that the gluing parameter p, for which the above gluing construction gives rise to an embedding into a trajectory space, if p is large enough, now also determines the target space Mxh"'Y itself. The result is stated as co, x'Y

j.L(x a ) = j.L(x{3) = j.L(x"Y) ,

"y

Proposition 2.60 Given a compact set of broken h-trajectories

K C M h"f3 X M hf3 'Y

XQ,X"

there is a lower parameter bound exists,

XI3,Xry

,

Ho, such that for any R

~

Ho a homotopy

r h"'Y(R) ': : f"Y ~

#n .

Xo:,X/3

XI3,X-y

Xo.,X')'

is a one-to-one correspondence of finite sets.

Proof. According to the above result it is sufficient to verify surjectivity. Let ua"Y be an h-trajectory from M~:~~~). Then, consideration of the solutions h"f3 hf3 'Y ua{3 E M x",xf3 and u{3"Y E M xf3 ,x'Y of the respective time-dependent gradient flow, which are determined uniquely by u a{3(O) = ua"Y( -R) and u{3"Y(O) = ua"Y(R) , leads us immediately to the broken trajectory forming the pre-image with re­ spect to #R' 0

together with a smooth mapping

# R·. K

there is an R > 0 such that the associated gluing map . M h"f3 x M hf3 'Y ~ Mh"'Y(R)

Mh"'Y(R)

XO!lX-y'

which represents an embedding.

2.5.3

Proof. For this case of the broken trajectories (u a{3, U{3"Y) we use again the original, 'symmetric' pre-gluing:

Let us now consider the manifold M~",Yf3 C [0,1] X p~~2,Yf3 of A-parametrized trajectories as a zero set with respect to the mapping

Ua{3(t (2.89)

(U a{3#R v {3"Y)(t)

=

+ R),

t

~

-1

jJ-[exp;} Ua{3]R (t), ItI ~ 1 eXPXf3 (+jJ+[exPX f3 V{3"Y]-R) { t ~ 1 v{3"Y (t - R) '

Gluing for A-Parametrized Trajectories

C a {3 : [0'1]X X pl,2 n ,YI3

(,\,u)

~

L 2 (pl,2 *TM) X ,Y/3

1-+

u+-1-VH a{3(A,.)ou

a

F

CHAPTER 2. THE TRAJECTORY SPACES

100

H°{3

Here, HD:!3 describes a A-homotopy h o ~ hI of regular Morse homotopies between fD: and f!3. In this framework, too, we wish to discuss the problem of a gluing operation combining such A-parametrized trajectories (A, u) with shifting-invariant trajectories of the gradient flow with respect to rand f!3, respectively, such that we again obtain A-trajectories (A', v) with A' generally differing from A. ~

~{3

Theorem 5 Let K c- M:o ,Y{3 x M£{3,z{3 be a compact set of mixed simply broken, A-parametrized trajectories (A, u>., v). Then there is a lower parameter bound Pi< and a smooth embedding

2.5. GLUING

101

In strict analogy with Definitions 2.48 and 2.49 we find a construction of a linear version of a pre-gluing map #x, this time for X = (A, u>.) #~, °v E [0,1] X p~~2,Z{3' Thus, we obtain a normal bundle L1- from Proposition 2.63 There is a lower parameter bound PI ~ Po such that the Fredholm operator DCD:!3(A, u>.#opv) ; lR x H~,2#ov --+ L; #ov is onto for >. p >. p all gluing parameters P ~ PI and mixed broken trajectories ((A,U>.),V) E R. Additionally, the composition of the linear version #x with the (lR x L2)_ projection onto ker DCD:!3(X) then induces an isomorphism: A..

'f'x =

#H : R x [Pi<' 00)

'----t

MH

(A,U>.,V,p)

f---4

(A, u>')#p v = (A, wj.)

X Q1 Z/3

HA

A Pn ) ---+ CI~c \ U>., v,

Wj. ( /I,

(

U>., vA)

and A(\ u>., v, Pn)

.lRxL~ roJker DG0{3 (x)

0

# x:

ker DCD:!3(\ u>.) x ker D v ~ ker DCD:!3(A, u>.#~v)

­

such that, provided any increasing sequence of parameters Pn weak convergence of the form

P

--+

--+

00, we obtain

A

As to the underlying analysis, we do not come upon anything new. We merely expand the Banach manifold, within which we are searching for the zeroes of a Fredholm map by means of the contraction principle, by the compact interval [0, 1] of A-parameters. Therefore, we will not provide a detailed proof at this stage. The strategy for proving this theorem is organized in two stages in a way which is absolutely analogous to the above investigations: At first, we have to define an appropriate pre-gluing map, in this situation similar to the time­ dependent case in Section 2.5.2, together with a suitable normal bundle L1-, which now lies within lR x H~,2 (p~~~Z{3 *TM). Finally, we must derive a unique smooth correction section within this normal bundle by means of a suitable contraction map.

As to the proof. Essentially, the proof of 2.50 remains the same up to expansion by lR. At this stage we wish to point to the trivialized version, which shall be mentioned again in the next chapter in relation to the discussion of induced coherent orientations. This will be treated in Proposition 3.14. Regarding a detailed proof in the trivial case together with the additional set of real parameters, the reader is also referred to [F-H]. This linear version of gluing in the trivial case is of the type lR X H ,2 X H I ,2 3 (7,~,() I

f---4

(7, f3=p' ~ + f3~p' (-2p) E lR x H I ,2 .

o Finally, considering the normal bundle L1- over R x [PI, 00), which is induced by this proposition in analogy with the former bundle L1-, and given a large enough lower parameter bound P2 ~ PI, we are again able to find a unique smooth correction section of (2.92)

'Y:

Rx

[P2, 00)

--+

L1­

'Y(A,U>.,v,p) = (7,~) E lR x H~,2#ov >. p

Definition 2.62 Referring to the corresponding pre-gluing map for mixed bro­ ken trajectories from Section 2.5.2 for an orientation, we now define the pre­ gluing operation #H,o : K x [Po, 00) --+ [0,1] x P;:,Z{3

such that

by (2.91)

represents the required gluing operation. Once again we observe exponential convergence (7,~) --+ (0,0) as the gluing parameter tends to infinity, P --+ 00.

(A,U>.)#~'Ov = (A,U>.#~V) ,

where we denote by

#~

the operation from (2.88), that is

(u>.#~V)(t) = {u>.(t),

V_2p(t),

t t

~ P- 1 ~ P+ 1

,etc.

(A, u>.) #~ v = (A + 7, exp,,>.#~v~) E

M: ,z{3 o

Chapter 3

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

WU(x) rh WS(y)

~

Mx,y

by the negative gradient field with some coherent orientation related to the critical points. Considering the framework of our analytical methods we need a con­ cept for such a coherent orientation of the trajectory spaces, which are treated as zero sets of the fundamental Fredholm map with respective endpoints. Ac­ tually, within this analytical context, coherence denotes the compatibility of the respective orientations with the cobordism relations from the compactness­ gluing-complementarity in the last chapter. In fact, the purpose of this chapter is to extend the former cobordism conc.ept for trajectory spaces to a concept of oriented cobordisms. The fundamental feature of this concept will be the no­ tion of orientations for Fredholm operators arising from the already well-known determinant bundle (see for instance [DonD. The reason for using this deter­ minant bundle for Fredholm operators may be explained as a generalization of the orientation of a manifold. Knowing that in our framework the tangent space of a trajectory manifold can be identified with the kernel of the surjective linearization of the fundamental Fredholm map, we wish to drop this regular­

--------------~~----

CHAPTER 3. ORIENTATION

104

ity assumption and define a substitute for the maximal exterior product of the tangent space, in which a non-vanishing element represents an orientation. This substitute concerning a non-surjective Fredholm operator consists of an appropriate combination of kernel and cokernel, so that a continuous variation of a thus oriented Fredholm operator preserves the orientation, whether the operator is onto or not. Hence we shall be provided with an orientation con­ cept for trajectory spaces including the regular trajectory manifolds as special cases.

II

--:---.---_::---~:-_._-----------_._----

.~':ail

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

3.1

Orientation and Gluing in the Trivial Case

3.1.1

The Determinant Bundle

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

Amax E

=

Adim E E

and A0 E = JR. ,

--- - -

3.1. ORIENTATION AND GLUING IN THE TRIVIAL CASE

105

we define the one-dimensional space of determinants with respect to E and F by

·.,;.1' . . .'· .·.

Det(E,F)

=

(AmaxE) 0 (AmaxF)* .

Now let H o and HI be ji:J;ed Banach spaces and let us denote by

F(Ho,Hd

c £(Ho, HI)

the open subset of Fredholm operators. Then we define the space of determi­ nants associated to such a Fredholm operator F E F(Ho, Hd as

Another point which seems worth mentioning concerns the topology of the underlying manifold M in the sense of the axiomatic homology theory in question. It appears to be founded in an abstract way within the existence and construction problem of such a coherent orientation for the Fredholm operators D u E Euo TM as u describes the trajectories of the negative gradient flow. Throughout the development of this analytical concept of a coherent orientation we keep in close analogy to the Floer homology in the symplectic case. The corresponding results can be found in [F-H]. At first, we shall review the construction of an orientation bundle for Fredholm operators in general. Then we shall verify the compatibility with the gluing calculus for Fredholm operators of the type F E E triv ' In the second half of the chapter we shall transfer the orientation concept from the trivialized framework to the manifold M, so that the topology enters when we ask for the existence of a coherent orientation. In contrast to the symplectic case in [F-H] we shall also consider non-orientable manifolds, whereby we shall need a slightly modified concept.

-_..._==-----------,. ------ - - - - - - -

Det(F) = Det(ker F, coker F) . If we further consider the continuous mapping of an arbitrary topological space X into the space of Fredholm operators ·l

~{

,~I;:

f :X

!~

~

F(Ho, Hd ,

we may define the space

U{x} x Det(J(x))

Det f =

xEX

Thus, the canonical projection map is endowed with the one-dimensional JR.­ vector spaces Det (J(x)) = 7[-l(X) as fibres.

The aim of this section is to show that the space Det f is a real line bundle on the topological space X. Then, the local trivializations uniquely determine the topology of this vector bundle. If we assumed dim ker f or dim coker f to be locally constant functions on X, I this result could be immediately concluded. By the following fundamental algebraic lemma we will derive the existence of the determinant bundle for general Fredholm operators without this assump­ tion. Lemma 3.2 Let o~

EI

dl dk_l ~ ... ~

E

k ~

0

be a sequence of finite-imensional JR.-vector spaces. Then there is a canonical isomorphism

Q9

¢: i

even

Q9 (A max E

(A max E i ) ---==-. i

i)

odd

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

_--7.-

CHAPTER 3. ORIENTATION

106

Proof. Let ell, ... , eI n1 be a basis of E I . Since d l is injective we may extend the linear independent vectors d l (ell),"" d l (eI n1 ) by e2}, ... , e2 n 2 to a basis of E2. Corresponding to this first step and due to exactness, we can find successively at each stage of the sequence an extension to a basis of the form

~_-

3.1. ORIENTATION AND GLUING IN THE TRIVIAL CASE

107

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

u

U(x) =X

XEX

di(eil), ... ,di (ein;),ei+II, ... ,ei+Ini+ll with

nk

= O. Then we define the isomorphism

for i=1, ... ,k-1,

¢ by

d;(ell) /\ /\ dl(elnJ /\ e21/\"'/\ e2n2 ) ( ®d3 (e31) /\ /\ e4n4 ® ... ® /\ e2j n2j ~

ell/\ ( /\e3I /\

yield a local trivialization of Detf on X. Actually, given (x,1/;,U(x)) and y E U(x), we have an exact sequence 0---. ker fey) ~ ker f,p(y) ~ R n ~ coker fey) ---. 0 ,

where

/\elnl®d2(e2I)/\ /\d2(e2n2)) /\ e3n3 ® ... ® ... /\ e2j'+1 n 2j '+1

dl(k)=(O,k) d2 (h, k) = h

Obviously, this isomorphism is independent of the special choices of the vectors of the bases, because any change in the bases operates by multiplication with the same determinant of the transformation on either side of the homomor­ phism ¢. 0 This lemma enables us to provide local trivializations of the de­ terminant bundle 7r : Det f ---+ X. (See the careful exposition in [F-H] for details.) Given x E X, we associate to the Fredholm operator f(x) a linear map 1/;; Rn ---+ HI, such that

(3.1)

l",(x):

Rn x H o (h,k)

---+

~

HI 1/;(h)

+ f(x)

.k

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

l",(y)

E

F(IR n x Ho, HI)

holds throughout U(x) together with the identity ind f",(y) = ind l1/J(x) for all y E U(x)

d3 (h) = [1/;(h)]R(JCY»)

This induces a natural isomorphism Det fey) ~ (Amaxker f,p(y)) ® (AmaxRn )* ,

Le. a natural isomorphism

Detf(y) ~ Detf.;;(y)

(3.2)

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

Det(f)

---+

X

is a real line bundle. Definition 3.3 The bundle Det f is called the determinant bundle of f and any non-vanishing section in this line bundle induces an orientation for the family f : X ---+ F(Ho, HI)

of Fredholm operators. exist.

We remark that such a section does not necessarily

Thus, the mapping

f,p(y):

Rn x H o

---+

R n x HI

~(o,f.;;(y)(h,k))

(h,k)

gives rise to a real line bundle Det foP on U(x), because we now deal with the constant functions dimker f,p(y)

and

dim coker f,p(y)

3.1.2

Gluing and Orientation for Fredholm Operators

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

--=--------=-------. - - - - - - - - - - - -

CHAPTER 3. ORIENTATION

108

analogous to the construction in the last chapter, we must restrict ourselves to asymptotically constant operators, that is, to K = + A K E ~triv satisfying

it

AK(t) = const

ItI

as

~ T for a T ~

°

Actually, referring to the equivalence relation

KA

'"

La

¢:}

A± = B±

and recalling that according to Lemma 2.15 these equivalence classes are topo­ logically contractible, we may choose an asymptotically constant representative from each equivalence class 8 K.

--+~

(p, FA)

f-+

FAp E e(K-,L+) AK(t+p), t~O { Adt - p), t ~ 0

Ap(t)

3.1. ORIENTATION AND GLUING IN THE TRIVIAL CASE

for all p ~ Po(K,L). Here, the lower parameter bound is determined by the asymptotical behaviour

+ P)I [-1,00] == Ai == AI: == Ad· -

P)l [-oo,IJ

109

ind 0# = +0 (ind,ind)

The crucial question now is whether there is a similar orientation concept for Fredholm operators as defined above, which is likewise naturally compatible with the gluing operation. At this stage it is important to state that, due to the contractibility of the equivalence classes [K] = 8(K+, K-), the associated determinant bundles

Det[K]

--+

8(K+,K-)

are in fact orientable.

o Det (L) ~Det(K#pL) .

Second, regarding the equivalence classes e K , 8L and 8K#pL, we have to verify that the orientation on 8K#pL induced by this isomorphism is indepen­ dent of the choice of the actual representatives. Once again, the fundamental Lemma 3.2, by which we may transform in a natural way the Fredholm oper­ ators K and L into the surjective operators K1/1 and £1/1, turns out to be the decisive utility. Thus let K and L be asymptotically constant operators from

E triv

AK(·

_. ,­

we can state the following relation:

Det(K)

Fp = FA(+p) ,

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

--------:;-_. __.. _ ._ _._'---..---------..- - - - - - - -

First, we require an isomorphism between the spaces of determinants associated to the respective Fredholm operators

Definition 3.4 Considering the isometric lR-action

lRxE

--_.,-------,--------------------

C

.c(H 1,2(lR, lR n ), L 2 (lR, lR n ))

with matching ends K+ = L -. We find a linear mapping

'l/J : lRk

for all p ~ Po .

L 2 (lR,lR n )

--+

,

such that the Fredholm operators

Remark According to the index theorem from Section 2.2, the Fredholm index operates additively under this gluing operation, i.e. ind (K#pL) = ind K

of equivalence classes [K] =

e K,

E/ '"

K E

~

and the map

ind : E --+ Z ,

lR k

X

Hl,2

(h,u) (h,u)

+ indL

holds for all asymptotically constant operators K, L E ~ with matching ends K+ = L-. Moreover, the equivalence class eK#L does not depend on the actual gluing parameter p and the respective choice of the representatives K E e K, LEe L. In a nutshell, if we consider the set

f; =

K1/1, L1/1:

--+

L2

f-+L·u+'l/J·h f-+K·u+'l/J·h

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

Det(K1/1) ~ Det(K)

and

Det(L1/1) ~ Det(L)

concerning the operators

K1/1' L1/1:

lRk x H 1 ,2 (a,u) (a,u)

--+ f-+ f-+

lR k X L 2 (0, L . u + 'l/J . a) (0, K . u + 'l/J. a)

-:-·6:±_X

=r

._-

_._ .._~~,_

;s"O=±S53±

e=~

§~t¥t'tttt.~t~¢ifii(t"':"'":~·,~·,~ =<

CHAPTER 3. ORIENTATION

110

We note that without loss of generality we may assume an R ~ supp(7P(a)) C [-R, R]

° satisfying

for all a E JRk ,

because the bounded linear mapping {3R:

L2

--t

L2

U

f---t

{3R'

u

associated to a smooth cut-off function (3R with compact support in [- R, R] gives rise to a continuous deformation satisfying ~

L(jR.1/J

.t(H1.2,L 2) ~

------>

L1/J

as R

__

"",~ "-~"'~"'::~=:'~.~~'''~:-'~-

''''L''~_ ~ ~~"'~~

~ ."",.,,,,-,-,,c"""""~

3.1. OPJENTATION AND GLUING IN THE TRIVIAL CASE

III

Proof. We observe that this proposition corresponds to Proposition 2.50 about the linear version of the pre-gluing of trajectories up to the finite-dimensional surjectivity extensions (IRk, 7P). Regarding these extensions with compact sup­ port supp 7P( a) C [- R, R], we may accomplish the proof by strict analogy. The reader is also referred to [F-H] . 0 Moreover, this 'surjectivation' by means of (IRk, 7P) comes to the core of the discussion of the trajectory gluing for the .A-parametrized gradient flow with respect to the orientation problem. There, too, the Fredholm operators D u become surjective generally at the earliest after extension by the partial derivative for the variational parameter .A. 3 Now considering the induced mapping

--t 00

,

K1/J#pL1/J : IRk and the surjective Fredholm operators form an open subset of the space of operators £(H I ,2, L 2 ). Given these surjectively extended Fredholm operators K1/J and we now define a gluing operation which is adapted to the situation:

L1/J,

X

JRk

X

H 1 ,2

--t

IRk

X

IRk

X

L2

(K1/J#pL1/J)(a,b,u) = (0,0, (K1/J#pL1/J)(a,b,u)) , which is equivalent to

K1/J#pL1/J

== (K#pL)1/J pffi1/J_p ,

Definition 3.5 Let Po = po(K, L) be the lower parameter bound associated to according to the above derivation of the determinant bundle, we are provided the asymptotically constant operators K and L, In order to take into consider­ with the natural ismomorphism ation the extension by 7P, we lift this lower bound to Pl = R + 1 + Po, and we (3.3) Det(K1/J#pL1/J) ---=--. Det(K#pL)

consequently define the mapping K1/J#pL1/J : JRk x JRk X H l ,2 --t L 2 by Lemma 3.2. Noting that the gluing Proposition 3.6 induces an isomorphism

(AmaXker K1/J) 0 (AmaXker L1/J) ---=--. Amaxker (K1/J#pL1/J)

for P ~ PI by (K1/J#pL1/J) (a, b, u)(t)

=

((K #pL) . u) (t)

+ 7P(a)(t + p) + 7P(b)(t -

p)

The central proposition with respect to this gluing of surjective Fredholm op­ erators may be stated now as

Proposition 3.6 Let Pp be the orthogonal projection in the Hilbert space JRk x JRk X L 2 onto the finite-dimensional subspace ker (K1/J#pL1/J). Then there is a lower parameter bound P2 ~ Pl such that the following holds for all gluing parameters P ~ P2 :

and resorting to the natural isomorphism (AmaxIRkr 0 (AmaxIR k)* ---=--. (Amax(IR k x IR k ))* we obtain an isomorphism for the associated determinant spaces of

(3.4)

Det(K1/J) 0 Det(L1/J) ---=--. Det(K1/J#pL1/J) ,

provided that the gluing parameter P is large enough. Finally, the natural isomorphisms yielded by the algebraic Lemma 3.2, which describe the trans­ formation of the determinant spaces with respect to the surjective extension, The Fredholm operator K1/J#pL1/J is surjective and the linear map­ i.e. Det(K1/J) ---=--. Det K, etc., ping

~: ~~x~~ ~ ker (K1/J#pL1/J) ((a,u),(b,v)) is an isomorphism.

f---t

Pp(a, b, up + v_ p)

lead us to the isomorphism for the original determinant spaces

(3.5)

Det K 0 Det L ---=--. Det( K # pL )

3See also Theorem 2 in Section 2.3.2.

CHAPTER 3. ORIENTATION

112

The next step is to verify that this isomorphism may be transferred to the determinant bundles of the associated equivalence classes e K, 8 L and 8K#L. In other words, we have to verify the coherence with respect to dif­ ferent representatives. Since the equivalence classes as topological spaces are T contractible within ~triv, we may choose connecting arcs K and LT with 1 O l T E [0,1] for each pair ofrepresentatives (K , K ) and (LO, L ), respectively. We are able to repeat the above analysis starting from these arcs instead of fixed operators together with the additional argument that the set of param­ eters [0,1] is compact and the arcs are continuous. Therefore, we once again find surjective extensions and corresponding isomorphisms uniformly with re­ spect to T or dependent on T. These isomorphisms give rise to a vector bundle isomorphism on [0,1], Le.

Det K T 0 Det F

"" --=--

(KT #p F )

\~/

(3.6)

[0,1]

for P ~ pO(K[O,l], L[O,ll). Hence we deduce that our orientation concept for the equivalence classes 8 K and 8 L is compatible with the gluing which yields 8K#L.

Before we finish this section by stating the result in a theorem, we wish to mention an important property. As we already noticed in the section on trajectory gluing, due to our construction, this gluing operation itself does not satisfy an associativity rule. However the mere orientation problem can actually be solved. Let K, L, M E ~triv be asymptotically constant operators with matching ends K+ = L - and L + = M-. Then, provided suitable gluing parameters PI,"" P4, the operators (K #Pl L)#P2 M as well as K #P4 (L#pg M ) are well-defined representatives of the same equivalence class 8(K-, M+). Thus we can construct a smooth line bundle E on [0,1] with the boundary fibers

Eo

= Det«K#PIL)#P2M)

and a vector bundle isomorphism

and

E1

= Det(K#p4(L#pg M ))

3.2. COHERENT ORIENTATION

113

[0,1] x (Det K 0 Det L 0 Det M)

~E ~

\~/ [0,1] such that <1>0 and <1>1 represent exactly the compositions of the gluing isomor­ phisms from Proposition 3.6 in the respective order. We sketch the proof. In principle, it resorts to the same analytical construction elements which we have used in our discussion about gluing up to this stage. At first, we define a gluing operation for three operators simultaneously with two free, independent gluing parameters. Then we have to prove the associated isomorphism property with respect to the surjectively extended Fredholm oper­ ators. Secondly, we have to show that, given gluing parameters that are chosen appropriately, we are able to homotope this isomorphism for three operators to the respective composition. The crucial step consists of finding a uniform estimate together with a homotopy to 0 for the difference term between the re­ spective isomorphisms as the gluing parameters become large enough. Finally we are able to sum up this section in the following theorem:

Theorem 6 Let K and L be asymptotically constant operators from ~triv with matching ends K+ = L - and let OK and 0L be orientations of the canonical determinant bundles on eK and 8 L , respectively. Then for all P ~ PoCK, L), the gluing operation #p: (K,L) f---'> K#pL induces an orientation OK#OL on 8K#pL = eK#L independently of p. This induced orientation does not depend on the actual choice of representatives K E e K and L E 8 L . Moreover, the associativity rule of is fulfilled, (OK#OL)#OM

3.2

=

OK#(OL#OM) .

Coherent Orientation

The outcome of the above analysis in the case of the trivial i-vector bundle i x IRn is an orientation concept which is supplied uniformly for all asymptot­ ically constant operators with coinciding ends. This means an orientation for

-_._---_.

114

CHAPTER 3. ORIENTATION

the equivalence classes eKE E, and it is compatible with the gluing operation. We now intend to transfer this result to the situation of the IR-bundles u*TM induced by the gradient trajectories u. Thus we principally have to consider the dependence on the curve u. Actually, this a priori issue touches on the topol­ ogy of the underlying manifold. The crucial observation will be that mainly the knowledge about asymptotical behaviour suffices and that we are able to orient all Fredholm operators uniformly as one having identical end terms and associated to curves with coinciding fixed endpoints. The only delicate point is the question if two given smooth curves u and v with identical endpoints u(±oo) = v(±oo) give rise to pull-back bundles u*TM and v*TM admitting trivializations coinciding appropriately at Tu(±oo)M = Tv(±oo)M. If this is guaranteed, we may transfer the results from the last section in a unique and canonical way. It is at this stage that we come upon the problem that not all two curves with coinciding endpoints satisfy this trivialization condition on the pull-back bundles in an a priori way. The obstruction of a common trivializa­ tion in the above sense is hidden in the topology of the connection to a loop u . v-I (provided a suitable reparametrization). Actually, the vanishing of the characteristic class WI (( U . v-I) *T M) is the necessary and sufficient condition for the trivializability of the bundle (u. v-I )*TM. In other words, we find si­ multaneous canonical orientations except for curves with coinciding endpoints, which close up to n-dimensional 'Mobius bands'. This topological obstruction is in fact an expression for the orientability of the underlying manifold M.

Nevertheless, a more detailed analysis will show that already the coincidence of the trivializations at one end proves sufficient, if we only can specify in a uniform way which end is the right one. Then, the possible dif­ ference at the other end, which may be expressed as a fixed reflection, turns out to be irrelevant. The only essential feature is the uniform and necessarily noncanonical specification for all curves, at which end we wish to have the iden­ tical trivialization. Although it might look unexpected or even cryptic to the reader, it is worth mentioning at this stage that this specification is in one-to­ one correspondence with the dimension axiom of the later homology theory and with the obstruction to the well-known Poincare duality in the case of closed but non-orientable manifolds. We shall come back to this issue in the section 'topology and coherent orientation' within the next chapter which is central to this monograph. Finally, having accomplished the transfer of the above notion of an orientation to the non-trivial framework of curves, we shall develop the con­ cept of a coherent orientation by means of the compatibility with the gluing operation for trajectories from the last chapter. This coherent orientation is the main feature which allows us to set up a homology theory with coefficients in Z.

__ ._­

3.2. COHERENT ORIENTATION

3.2.1

115

Orientation and Gluing on the Manifold M

We shall accomplish the orientation process for the Fredholm operators from EU*TM not only for the trajectories of a special gradient field but in general for all smooth compact curves u E COO (IR, M). Therefore, we consider this space of curves to be equipped with the Whitney C OO -topology.4 Since we merely deal with pull-back bundles u*TM on IR, Lemma 2.19 guarantees that the induced covariant derivation from T M is Fredholm admissible. Hence, we drop V' from the notation. Definition 3.7 We call two Fredholm operators K E

E u ' TM ,

L E

EV*TM

along supporting curves u, v E COO (IR, M) equivalent if the asymptotical iden­ tities u(±oo) = v(±oo) and K± = L± hold. Any pair of trivializations cPu:

u*TM ~ IR x lRn

'l/Jv:

v*TM ~ R x lR n

is called admissible for such equivalent operators (u, K) '" (v, L), if the identity cPu(-oo)= 'l/Jv(-oo) holds at the identical lower ends of the curves, the relation

±1 1

cPu(+oo)· 'l/Jv(+OO)-1 =

1)

E

GL(n,lR)

( at the identical upper ends and if the equivalence defined above for the trivial framework cPu(K)

= cPuK cP:;;1

"'triv

'l/JvL'I/J;;1

= 'l/Jv(L)

is given. Let us denote the equivalence class of (u, K) by [( u, K)] or merely by [K-, K+], as it is already determined uniquely by the endpoint operators K± E End(TM).

Remark Given two curves u and v, which are connected to a trivializable loop with respect to a suitable reparametrization, Le. WI ((u. v-I)*TM) = 0, we always find an admissible pair (cPu, 'l/Jv) associated to (u, K) '" (v, L), so that

R

4Relating to the differentiable structure on introduced in the first chapter, we note that weak and strong Whitney topology on COO(R, M) fall together.

,.,1

.:-'-"

4'>:',· "" .::::~-

n..

'""'-

--

0..

"_.r.

"'"'_,,"'-"'" '

-::-;z~,;,;

- --

"Ow

CHAPTER 3. ORIENTATION

116

,~~;;;";;~'~:'i!~~, .;<".,,,,.

~" ..?~:~~j~~;fr~it'f~~~~:~ii~~iE',;;rt2Zil~~'$;~:~;.::'=i;;~::~:::~~;*:=

~'"i~~'tt~Jc~~=~~:;;;:~Jk~;I~r;,~$~-;'~-==':;::;-,ii";'~"::i;~;~-S-~1;~":~~,;;;;:~~~~~

3.2. COHERENT ORIENTATION

117

the identity 4>u(±oo) = 'If\(±oo) holds. Otherwise, if u . V-I yields an n­ Considering the determinant bundle Det e K on e K together with the orien­ dimensional 'Mobius band', we choose asymptotical trivializations C(±oo): tation induced by OK, we are provided with an orientation 'l/J (OK) for Tu(±oo)M ~ ffi.n, such that

'l/J(K) = 'l/J. K· 'l/J-l

C(±oo)K±C(±OO)-1 = C(±oo)L±C(±OO)-1

+1)

(

Hl,2(~) ~ Hl,2(~) and resp. L2(~) ~ L2(~)

which arise from 'l/J and which are again denoted by'l/J.

+1

S=

Then this new orientation of the determinant bundle is compatible with the original one, that is 'l/J(OK)

thus yielding C8 (+00) satisfying

8K

via the Hilbert space isomorphisms

has a diagonal shape. Then, we compose C(+oo) with the reflection -1

E

~

OK .

= S· C( +(0). Hence, there are trivializations 4>u, 'l/Jv 4>... (-00) = 'l/Jv(-oo) = C(-oo)

and

4>... (+00) = C(+oo), 'l/Jv(+oo) = C 8 (+00) or vice versa 4>... (+00) = C 8 (+00), 'l/Jv(+oo) = C(+oo), and which form an admissible pair by construction. So, to sum up, there is always an admissible pair of trivializations associated to (u, K) '" (v, L). The next lemma shows how to orient such equivalent operators si­ multaneously in a unique way by means of these trivializations.

Lemma 3.8 Let (u, K) '" (v, L) be equivalent Fredholm operators and let (4), 'l/J) and (4)', 'l/J') be respectively admissible pairs of trivializations. Moreover, let DetK and DetL be oriented by OK and OL· Then the pair (4), 'l/J) induces compatible orientations 4>(OK) ~ 'l/J(od on the determinant bundle of the trivialized class 8K-1 = 8,pL'.p-l if and only if the same is true for the pair (4)', 'l/J') on e'K,-l = 8,p'L,p,-1, that is 4>'(OK) ~ 'l/J'(od . Let us state separately the decisive step of the proof of this lemma as Auxiliary Proposition 3.9 Let ~ be a smooth ~-vector bundle and OK a fixed orientation for K E E~. We denote the set of sections 'l/J E Coo (End (0), which are pointwise invertible, by GL~. Further, let us choose any section 'l/J in GL~ satisfying the additional asymptotical condition 'l/J(±oo)

= id~(±oo)

.

Proof. Resorting to the structure group of ~, we reduce the given operator K E E~ and the transformation 'l/J successively to a form which can be analysed easily and explicitly. At first, we observe that we may start without loss of generality from the trivial bundle ~ = iR x ffi.n. Namely, if we choose' any trivialization 4> : ~ --t ffi. x ffi.n , 'l/J and K transform as 'l/Jtriv

= 4>. 'l/J. 4>-1

E

GL triv

and

'l/Jtriv(Ktriv )

= 'l/J(K)triv

Provided that the assertion is true for the trivial bundle, that is ('l/J (OK))triv

= 'l/Jtriv(OK,riJ ~ OK'dv

,

we can transfer this orientation equivalence to the original bundle by means of 4>, because it holds that

OK

~

0L

if and only if 4>(OK)

~

4>(od .

The next step of the proof consists of showing that it is sufficient to verify the equivalence 'l/J(od ~ 0L

for any Fredholm operator L i.e. there is a continuous arc

Lr

:

[0,1]

--t

E

F(H 1 ,2, L 2 ) which is homotopical with K,

F(H 1 ,2, L 2) satisfying

L o = K, L 1 = L .

Such an arc would then yield the equivalence

'l/J(OK) ~ 'l/J(OL) ~ OL ~ OK due to the vector bundle isomorphism

-=-c~

---_.

~ ~_='-.~=::'C="=-'-._~=-'~~~~~~~~~~~~ . . .~ ~.. -~ -=---:::~~~~~~~~~~2~~-~='~--=-~ ~~~~ 3.2. COHERENT ORIENTATION CHAPTER 3. ORIENTATION 118

Det(L r ) ~Det(7jJ(Lr))

a + II

K = at

[0,1]

E GL triv

with

7jJ(+00) = ¢(-oo)

7jJ(K)

with

= 7jJ. K· 7jJ-1 =

= 7jJ(K)#¢(K)

At

= A K, Ak = AM and At = II .

Let us assume for the moment that we have already proven the assertion for the special case (3.8) N E I: triv with A~ = II .

a

-

o(L)#o(K)#o(M)

~ o(L#K#M) (3~) o((ll#7jJ#ll)(L#K#M)) '"

o

o

K+tl.,p.

Regarding the difference term we calculate the estimate

tl.,p

representing a compact perturbation of K,

Iltl.,pIl.c(Hl,2,L2) :::; Ilepllo,2 < 00 .

(3.9) T

> 0, we define

t 7jJr(t) = 7jJ( -) ,

so that the estimate (3.9) amounts to

Iltl,pr II

(3.10) Therefore, provided that arc

TO

o(L)#o(7jJ(K))#o(M).

~

1

- . const T

> 0 has been chosen large enough, the continuous

K;o:

(0,1] T

Then, due to the gluing operation, the general assertion is concluded from

(3.7)

-1

T

Since our orientation concept for I:triv is compatible with gluing, we now choose asymptotically constant operators L, M E I: triv satisfying

ll,

+ (7jJ(7jJ:-1) + ll)

at+ll+ep.

K(+oo)=K(-oo) ,

(7jJ#¢)(K#K)

AL =

:t

o

we obtain the identity

(3.7)

o

o

o

Given any

and

K,K

sin
cos\p(t)

and 7jJ(t) =

appropriately. Since we know from the assumption that 7jJ is endowed with the end operators 7jJ(±00) = II E GL+(n, JR.)

7jJ,7jJ

cos \P(t)

- sin \P(t)

= COO(i,GL(n,JR.))

and that 80(n, JR.) is a deformation retract of GL+(n, JR.), we are able to find a continuous path 7jJr from 7jJ to a 7jJo E Coo (JR, 80(n, JR.)). Moreover, by means of a continuous asymptotical reparametrization, we may find a continuous arc ending in an asymptotically constant 7jJ. Regarding such a 7jJ E GL triv , we can construct a similar gluing operation, so that, given

119

where r.p E Coo (i, 8 1 ) is asymptotically constant. We obtain the representation

We shall now simplify the section

7jJ

-~~¥~O

Considering all these reduction steps, we may start without loss of generality from the special situation

\ffi/ induced by 7jJ. Hence, we may assume without loss of generality that the op­ erator K E I: triv is asymptotically constant and that it is admissible with respect to the gluing operation introduced in the last section.

_ _ _ ~~

---'>F(H 1,2,L 2 ) ~

K

+ Ttl.,pro

runs merely through isomorphism operators H 1 ,2 ~ L 2 , because we already know that K

=

:t

+ II : H 1 ,2 ~ L 2

---­ ==

::f~''''''''~-"-'''''_~:''''3'''-

CHAPTER 3. ORIENTATION

120

describes an isomorphism. Finally, we see that on one side the transformation 7/Jr o maps the orientation [101*] of K onto the orientation [101 *] of 7/Jro K 7/J;:;/ . On the other side, we can deduce from the homotopy K~o that both orientations correspond to each other by continuation. Hence, we obtain the relation (3.11 )

o This result enables us to prove Lemma 3.8.

Proof of Lemma 3.8. Let us start from the orientation equivalence O(c/JKc/J-l) = c/J(OK) ~ 7/J(od = O(7/JL7/J-l)

Then we have to verify the equivalence of the orientations

(3.13)c/J'(oK)

= c/J'c/J-l(o(c/JKc/J-l))

and 7/J'(o£)

= 7/J'7/J-l(o(7/JL7/J-l))

Moreover, on the one hand, we assume the asymptotical identities c/J( -00) = 7/J(-00) and c/J'(-oo) = 7/J'(-oo), so that (7/J7/J,-lc/J'c/J-l)(-oo) =

n E GL(n,lR)

Since on the other hand c/J' c/J-l and 7/J7/J,-l describe curves within GL(n, 1R), the inequality det(7/J7/J'-lc/J'c/J-l)(+oo) > 0 follows. Thus, we are led to the identity (7/J7/J,-lc/J'c/J-l)(+oo) = n E GL(n,lR) . Altogether, we are able to apply the auxiliary proposition and hence derive the equivalence (3.14) c/J(OK) ~ (7/J7/J'-lc/J'c/J-l)(c/J(OK)) . To sum up, the assumption (3.12) implies the assertion (3.13).

0

By means of this last lemma, we now are able to transfer the ori­ entation concept from the trivial bundle as treated in the first section to the Fredholm operators {(u,K) IKE ~u·TM} in relation with the above equivalence relation:

~

- ",ii,·

.....""~>:it"....,."""~ .

3.2. COHERENT ORIENTATION

121

Definition 3.10 Any orientation of an L E [(u, K)J induces a unique orien­ tation of the equivalence class [(u, K)], that is to say that given K '" L the equivalence OK ~ 0L holds exactly if c/J(OK) ~ 7/J(OL)

holds for any admissible pair of trivializations (c/J, 7/J).

OK ~ 0'I/Jro (K) = 7/Jro (OK) .

Regarding the continuous homotopy with respect to the variation of r E [1, roJ, we are led similarly te the correspondence of the orientations 7/Jr o(OK) and 7/J(OK) in the determinant bundle Det 7/Jr(K) on [1, roJ. From this we conclude the assertion 7/J(OK) ~ OK .

(3.12)

2::;"''':;;'';';'-''~''''"''''''''''''''''''''-

Remar k If we defined the equivalence classes to be of smaller a priori extent by the condition that trivializations with coincidence at both ends should exist, we obviously would be provided with more classes to be oriented coherently in the case of a non-orientable manifold. As we shall see from later features within the construction of a Morse homology, however, we do not have that much freedom regarding the choice of orientations for operators on curves which may be linked together yielding a 'Mobius band'. Actually, the dimension axiom will give rise to a condition on coupling between the orientations of two such classes. The crucial point worth mentioning is that we have already built in this condition by demanding the coincidence of trivializations at the negative end within Definition 3.7 of equivalence which might have seemed rather unmotivated to the reader. Assuming that we had fixed the coincidence at the positive end of the curve, we would be led to the 'wrong' dimension axiom, as shall be illustrated below in Section 4.1.4 on the example of JP>2(1R). Having accomplished the transfer to the non-trivial framework of varying curves, we now introduce a similar corresponding gluing operation by the trivialization method. We have to verify that this gluing is compatible with the trajectory gluing as regards the orientation induction by gluing. Definition 3.11 First, we once again define the pre-gluing operation from the last chapter for arbitrary curves u, v E coc> (ii, M) satisfying u( +00) = v( -00). Given P ~ po(u, v) we define

u(t + p),

u#~v(t) =

t~ ~-1

j3-(t).exp;;l (u(t+p)) , It I ~ 1 expy (+j3+(t). exp;;l (v(t _ P))) { t~1 . v(t - P) ,

Here, the lower parameter bound Po (u, v) is again derived from the asymptotical condition u(t + Po), v(t - Po)

E

U(y) . for t> -1 and t < 1 respectively

in relation with the normal neighbourhood U(y) of the exponential map. Let us now consider trivializations c/J:

u*TM

~iixlRn

7/J:

v*TM

~ iR x IR n

CHAPTER 3. ORIENTATION

122

3.2. COHERENT ORIENTATION Thus, defining

by starting first from a smooth trivialization

f: TMIU(y)

---=-. U(y)

123

K¢#~L",

x JRn

= K¢as#pL",as

for

p ~ Po

=1

,

we obtain an orientation of K ¢#~L", induced by #~ in a way which is inde­ on the neighbourhood U(y). This induces immediately i:-smooth trivializa­ pendent of the construction of K¢as and L",as based respectively on K¢ and tions on the pull-back bundles ul (-l,oo(TM and vI [-oo,l)*TM, which may be L"" due to the homotopy invariance described in the last section. Moreover, extended to trivializations ¢ and 'l/J, such that the identities since there is a T ~ 0 such that f1u«-1,00j) == ¢U-l,oo], f1v([-00,1)) == 'l/JI [-00,1) K¢#~L",(t) = { (K¢)p(t), t ~ -T are satisfied. Thus we are able to define a gluing operation for trivializations (L",)_p(t), t ~ T in a compatible way: holds for all p ~ Po, the class 8K.p#~L", is also independent of the noncanon­ ¢#~'l/J : (u#~v )*TM ---=-. iR x JRn ical construction of K ¢ as and L", as' Finally, we may transform this glUing construction back to the non-trivial situation by means of the analogously in­ ¢(t + p), t ~ -1 troduced trivialization ¢#~'l/J, ¢#~'l/J(t) = f u#~v(t), It I ~ 1 { 'l/J(t-p), t~l K#~L = (¢#~'l/J)-I(K¢#~L",)(¢#~'l/J) E E(u#~v)*TM . By means of these trivializations we shall transfer the gluing operation for E triv to the pull-back R-bundles on the curves, but first, we have to introduce the condition of asymptotical constance.

Definition 3.12 Given K E K¢

= ¢K¢-l

and L E

Eu*TM

and

L",

Ev*TM,

= 'l/JL'l/J-l

We notice again that, due to homotopy invariance, the class [u#~v, K #~L] and the associated orientation induced by #~ from o([u, K]) and o([v, L]) are independent of all the above noncanonical construction elements. Summing up, we consider the noncanonical gluing operation

we obtain

#0 :

E E triv

=

[u#~v,K-, L+] = [u,K±]#[v,L±]

which induces a unique and canonical mapping of orientations

(K¢)p = (Kp)¢p ,

(¢)p

-+

(u,K)#~(v,L) = (u#~v,K#~L) ,

With regard to the time-shifting we notice the identity

because the JR-aetion commutes with the partial derivative

[u,K-, K+] x [v,L-, L+]

(o[u, K], o[v, L])

ft,

f-7

o[u, K]#o[v, L]

with regard to the determinant bundles

(¢p) .

Det ([u, K]) -+ [u, K], etc. The next step is to define the halfway asymptotically constant operator associ­

Due to the construction, the homotopy invariance and particularly the associa­ ated to K¢

tivity may be deduced from the former trivial framework: K¢as = :t + /r· AK.p +,8+. Ak.p (3.15) (o[u, K] # o[v, L] ) # o[m, M] = o[u, K] # ( o[v, L] # o[m, M]) . and L", as analogously, having in mind the identity AL,p == AI.p' Obviously, the following relations are true: We shall round off this section by verifying that this orientation

K¢ ~ K¢as

and

L", ~ L",as

in

8K.p and 8L""

respectively.

Hence, the gluing operation already known on the trivial bundle for the operator K¢as #pL",as induces the isomorphism Det K ¢ 8IJ is> Det L", as

---=-. Det (K ¢ as #L", as)

"

concept in relation to the gluing of arbitrary smooth curves u E COO(iR, M) and Fredholm operators K E Eu*TM includes our former trajectories as a special case. Along these trajectories we consider the surjective linearizations in local coordinates D u = DFu(O) : H 1 ,2(U*TM)

-+

L 2(u*TM), u E Mx,y C C'::v

124

CHAPTER 3. ORIENTATION

3.2. COHERENT ORIENTATION

that is, the differentials of

Fu = '\7 2 exp~l O(:t

for all

+ '\7 f)

0

expu : H 1 ,2(u*D)

---+

E

7

L 2(u*TM) .

[0,1]. As a consequence, the isomorphism 2

·L D#p = Dh 1 = ProJker Dhl

In other words, the gluing #0 defined above is compatible with trajectory gluing as far as the orientation induction is concerned.

induces the same orientation on

Lemma 3.13 Regarding the gluing operation for time-independent trajectories

as

M;,y x My,z

3

(u, v) ........ u#pv

E

.L2 P rOJker Dho

Mx,z

---=-. ker Du#pv


on ~

Additionally, the identity of the classes

X

ker Du,triv x ker Dv,triv

K

holds.

lK(u, v, p)

(~,

C~A~, M) ,

........ expu#~v

(U,V,7)

c P;,';,

(7 'IK(U, v, p))

yielding the homotopy h T = H(·, 7) from #~ to #p. Provided that the gluing parameter p has been chosen large enough, the differential ---+

p

,

PrOjt:rDu#~V.triJB-~p + (3+(-p)

= Du,triv

and L

= Dv,triv

()

~ ker Kas#pL as L2

........ Projker (~p

+ (-p)

which similarly goes merely through isomorphisms. This is in fact possible, because the compact interval of time around 0, on which the curves A K88 #pL • a and AD #0 triv from COO(~, End(~n)) differ, decreases as the gluing parameter increases. The central element of this homotopy through isomorphisms is given by a continuation of the form U

DhT ( u, v) : ker D u x ker D v

---=-. ker Du#ov triv ........

ker K as x ker Las

Let us now consider ---+

D#op

throughout the remainder of the proof. It remains to show that we can ho­ motope this trivialized version
Proof. Let us assume any fixed pair of trajectories together with compact neighbourhoods N(u) c Mx,y and N(v) c My,z' Given the compact set K = N(u) x N(v) we choose a fixed gluing parameter p ~ p(K). Then the glued trajectory is represented as

K x [0,1]

0

~n. We denote the operators from E triv by

[u#pv, Du#pv] = [u#~v, Du#~Dv]

H:

2 Dh 0 = P rOJker ·L D#~

(~, ()

o[u, Du]#o[v, D v ]

= expu#~v

0

According to the construction of the pre-gluing map in the last chapter we can find admissible trivializations of the pull-back bundles u*TM and v*TM, such that we are led to the linear mapping

induces the same orientation o[u#pv, Du#pv] as we obtain from

u#pv

Dh 1

0

[u#pv, Du#pv] = [u#~v, Du#~v]

endowed with fixed orientations o[u, D u ] and o[v, DvJ, we observe that the isomorphism D#p : ker D u x ker D v

125

p'Ul

7 ........ [(1-7)(3-+7] ~p + [(1-7)(3++7]

H 1 ,2 (h T (u, v )*TM)

proves injective for all 7 E [0, 1]. This follows from the deductions in the last chapter in the same way as we obtain the surjectivity of DhT(u,v) and the isomorphism relation

(_p'

Actually, the proof that the corresponding linear gluing version is an isomor­ phism may be accomplished according to the scheme of Proposition 2.50 for each parameter 7. Hence we conclude the proof of the lemma. 0 I!I

(3.16)

·L 2

ProJker DhT

0

DhT

:

~

ker D u x ker D v - > ker D hT

5Note that the surjective operators form an open subset such that suitable asymptotically constant operators can be found.

126

CHAPTER 3. ORIENTATION

Remark Similarly, we may state a corresponding result for the gluing operation for mixed broken trajectories, now based on a linear gluing version of the type (~,(,p)~f3=p·~+f3~p·(-2p and

-+

= l' + _I_VHQ;(3 0 'Y

. _ .• _ .. _ __,. . _>'. •• .... .

_

~

·

'

"

<

'

"

·

"

'

,

·

.

~

~

""00

,w_.~.~,

,

,

~

~

,

,

~

~

~

_

,

~

~

_

~

·

'

~

M

"

,

"

~

,

~

.

"

~

'

_

n

'

"

~

_._,."._~·,,_._,._.

,

._._._•.__ ,..• __ ., .__ . .• __•.•. __ ,...• __ ... ..·".-..

"'"'"_.,_._.",_.....,_,,..__', ...

'

.• ,, ,~_

_

•.

__

.

~

,

~

,

'

"

·

~._"_~".,

__

~

._,

__

~.""'_""

~

..

__

'

~

~

·

_

~

,

·

_

·

.

.

,

~

,

,,~~~,_'"_

_

-.,,__._ •.•.• ,•.._.•

••

_

~

,

~

;

"

"

~

•.

.

.'. ~

"

~

.

~

•

,,,~_,,,.,,_,,~,,_.~,,

~

w

·

~

~

~

,

,

~

·

,

~

_

'

,

,

.

_

~

"

"

~

·

,

~

'

.

w

~

'

,

~

_

.

~

_

~

~

<

~

<

.

_,," __",_•.__."... _,..

127

d3 (T) = [D 1GQ;(3.

T]R(D

2

Ga 13 )

AmaXker D 2GQ;(3(A, u>.) ® T>.[O, 1] S!'

Amaxker DGQ;(3(A, u>.) ® Amaxcoker D 2GQ;(3(A, u>.)

--=-. Amaxker DGQ;(3 on the components of the parameter-trajectory spaces M:!.. Thus, the choice of

the fixed orientation !!>. == 1 on T* [0, 1] ~ IR gives rise to natural isomorphisms of the above line bundles on these manifolds of A-parametrized trajectories (3.19)

As we know from the transversality results, DGQ;(3(A, u>.) proves surjective for a generic metric. The Fredholm operator D u>. = D 2GQ;(3(A, u>.) E ~u'TM with the index J.L(xQ;) - J.L(y(3), however, does not need to be surjective at all. This can merely be provided as an a priori statement for the boundary parameters A = 0,1. Hence we conclude that DGQ;(3 (A, u>.) is a Fredholm operator with index J.L(xQ;) - J.L(Y(3) + 1 for all (A, u>.) E M~'Y/3' We now wish to compare the gluing embedding

f/3 f" H ~ H #p : M xc"Y/3 x M Y/3,z/3' MXa,Ya x M ya ,z/3 :) K,---; M Xa ,z/3 ,resp.,

H

for A-parametrized trajectories with the more general gluing of Fredholm clas­ ses with regard to the orientation induction. At this point, we once again have to resort to the fundamental sequence Lemma 3.2. Dealing with the orientation problem is crucially simplified by the determinant bundle and the gluing with respect to classes of Fredholm operators [K] always being led back to the respective surjectively extended operator K,p : IRk x H 1 ,2 -+ L2 as far as the concrete construction is concerned. But this exactly concerns the relation between the operators DGQ;(3 : IR x H 1 ,2 -+ L 2 and D2G(~(3 : H 1 ,2 -+ L2. Let us consider the following short exact sequence: ker D 2 GQ;(3 ~ ker DGD: 11 ~ T>.[O, 1] ~ coker D 2 GQ;(3

.•. __••'._'_'

., .• , "

Det D 2GQ;(3 ® T* [0, 1]

DGQ;(3(A, u>.) : T>.[O,l] x H 1 ,2(u*TM) -+ L 2(u*TM)

(T,~) ~ D 1GQ;(3(A, u>.)· T + D 2GQ;(3 (A, u>.)· ~ .

-+

__

for all (A, u>.) EM:!.. This amounts to a canonical bundle isomorphism

associated to the A-homotopy HQ;(3 : [0,1] x M -+ lR. Given the local coor­ dinates (expu, V 2 expu) at (A, u>.) E M?!an Y/3 = GQ;(3-\O), we consider the corresponding linearization

(3.17) 0

• •__•

"".,~"

dl(~) = (O,~)

-+

F

H

...

d2(T,~)=T

(3.18)

L 2(pl,2 *TM) be the mapping Xal~

GQ;(3 (A, 'Y)

~_

According to the fundamental Lemma 3.2, it induces a canonical isomorphism

Orientation and Gluing for A-Parametrized Trajectories XQ'~

,-_,~.,,.

where

f3;·6 p +f3t·(, respectively.

(~,(,p)~f3;p·~(1+T)P+f3:p·((-1+T)P'TE[-l,l].

pl,2

"" ,,, __

3.2. CCHERENT ORIENTATION

We can reduce this result to Lemma 3.13 by means of a continuous variation of the form

Let GQ;(3: [0 , 1] x

~

_

~

"

._,.,,.,__• __ ,_.,'_.. _.,.__ >.,,~

~

,

0' ~

·

,

,

~

_

.

,,,,u

~,<

.

,

,

=

,

,

~

_

~

_

~

~

~

~

"

~

·

~

"

_

~

~

.

_

'

.

"

."., , ••.._, __

_

."._,_""" __ ,,.'._ •••• _•••

,.~'~."._.'._""_~~_'"_m

~

_

~

~

~

~

•• _,_ .•

.. __

~

,,~~,~,_

~

"

~

,

_

,

~

~

_

'

·

, , ' _ _ • ".

o

G

V

,

.

~

~

_

.

_

~

.,_"' __ ",·••

_

-

... P, '

_~'

'

....

'

.~

'

·

,

~

_

·

,

n

~

_

.•

.

. _..• ,._._----_._-_••__ .0-." .•-.'-_._--',.,- •.__•.,._- ... __ •

__.•_""".__,__

.,,_,,~.~._,

,_ _

'

.__....

_ _,__,_ _

-

~

o

.,_._.•

~~,

. _ ._ _ _ _...

,

••._.__.•.

• _ _•._ " " .__

~

••

~

._._••._.

~

,."••.•_.,

""._._._._".

_

•

~

.•

~

.

__. _ .

~

••• ,_~_._u

~

.__•

,

•

t

•

f

• __,

_ _"

-

._,

-

c"

.

•__••".

-

._._. ._~

-

_.__

.

.•

.

_ _ _ _ _'"

.

_ _ _ _ _ _.

_ _. _ _ _ _ _ _

-+

a

n : Det D 2GQ;(3 --=-. Amaxker DGQ;(3

.

Given any pair (A,U>.) E M?!a,Y/3' n uniformly induces an identification of an orientation of the operator D u >. E ~'L'TM with an orientation of the tangent >. space T(>.,u>.)M?!a ,Y/3'

Proposition 3.14 Let us consider the gluing operation for A-parametrized tra­ jectories H H M xa ,Y/3 x M Yf/3· /3,z/3:3 (( A, u>. ),v) ~ (A, u>. ) #pH v -_ (A, w>._) E M Xa ,zl3

together with given orientations o[u>., DuJ and o[v, D v ]. Then, the isomor­ phism H H f/3 D#p : T(>.,u>.)M xa ,Y/3 x TvM Y/3,z/3

S!'

-+

T(>.,u>.)#:v

M

H

X a,Z/3

in connection with the isomorphisms n(A,U>.)

Det·D u>.

n((A, u>.) #;;v)

Det D w >-.

AmaxT.(>',u>.) M HX ,Y/3' a H S!' AmaxT. -+ (>.,u>.) vM Xa ,Z/3 9;

-+

#:

induces the same orientation o[w):., D w >-.] as we obtain from the gluing o[u>., DuJ # o[v, D u ]. Here we use the identity of the classes of glued operators [w):., D w >-.]

=

[u>.#~v,Du>.#~Dv] .

CHAPTER 3. ORIENTATION

128

Proof. By analogy with the proof of Lemma 3.13 we regard homotopies and trivializations at suitably large gluing parameters, so that we can reduce the assertion to the following special case. Let K, L E E triv be halfway asymptotically constant with matching ends K+ = L - . We further assume that L is already surjective and that 1/J : JR ---+ L 2 is a linear mapping, such that

K,p : JR

X

H 1 ,2

---+

L2

(T, ~)

f---t

1/J. T + K . ~

with

H 1 ,2 ---+

L 2 by

AK(t), { Adt-2p),

AK#~dt)

t ~ P

(3.20)

ffixL 2

Proj~

f---t

~(l-T)p

f---t

K",#J!L

+ ((-l-T)p,

{}p: ker K,p x ker L

((a, u), v)

9!!

(a,~+(-2p)

T

Pp(a, up + v_ p)

f---t

and compare it to the linear gluing version (3.22)

Affiaxker L ~ Affiaxker K,p # pL

induced by {} p, we observe that due to the identities ker K,p

L2

K,p : JR x H 1 ,2

---+

JR

(a,u)

f---t

(O,K,p.(a,u))

X

Affiaxker K,p 0 JR'

= Det K,p

---+

Det K

is expressed exactly by the natural isomorphisms defined above,

n- 1 :

Affiaxker K,p ~ Det K ,

the proof is accomplished.

0

Coherent Orientation Up to now we have developed a concept which enables us to join Fredholm operators K E Eu"TM along compact curves u E COO(i:, M) into equivalence classes, which may be oriented as one with consideration of the gluing operation. The last step toward the key feature of our intended homology theory consists of uniformly orienting all those operator classes in a way which is compatible with gluing, i.e. in a coherent way.

A = { [u, K]

used in Proposition 3.6. Considering the isomorphism fO> '
'd01"

'---+

Definition 3.15 Let

¢p: ker K,p x ker L,p ~ ker (K,p#pL,p) Affiaxker K,p

= Affiaxker K,p

E [0,1] ,

ker (K,p#pL)

---+

Det ¢p : Det K,p 0 Det L,p ~ Det (K #pL ),ppffi,p-p

~

we are able to transform the actual linear version of gluing into the shape used in Proposition 3.6 for any lower parameter bound Pi that is large enough. Therefore, without loss of generality, we may start from the version (3.21 )

--=-. Det (K #pL ),ppEbO

induced by {} p by analogy with the deductions in Section 3.1.2. On the other hand, ¢p yields the isomorphism

Affiaxker K,p

is an isomorphism. The proof is accomplished by exactly the same steps as are presented in [F-H] for the more general version of the gluing of asymptotically constant Fredholm operators. By means of the homotopy of the form (~, ( )

Det {} p : Det K,p 0 Det L

morphism Det K 0 Det L ~ Det (K #pL). Since the composition

~ ker (K,p#Ij L)

(a,~,()

Thus on the one hand, the choice of the canonical basis l' E JR' gives rise to the isomorphism

Using the natural isomorphisms Det K ~ Det K,p, Det L ~ Det L,p, etc., we are led to the result that Det {} p as well as Det ¢p induces the same iso­

t~p

and choosing p large enough, we deduce that

{}p: ker K,p x ker L

Affiaxker K,p 0 AffiaJ
t,

(3.24)

1/J . a + K #;L . ~

129

and ker K,p#pL = ker (K#pL),ppEbo we may also state it as

(3.23)

is surjective, too. Then, defining K,p#Ij L : JR x

(K,p#: L)(a,~)

3.2. COHERENT ORIENTATION

= ker K,p for

I u E COO(i:, M),

K E EU"TM

}

be the set of the equivalence classes of Fredholm operators along smooth compact curves. Then any map a, which associates an orientation a[u, K] of the corresponding determinant bundle Det([u, KJ) ---+ [u, K] to each class [u, K] E A, is called a coherent orientation, if and only if it is compatible with the gluing operation

a[u, K]#a[v, L] = a[u#v, K #L]

====,....., .- "--

------.-

---";--=---===,-~=."======",,==--====---=:::,==,,"====,,,=.=,,,,-,==,,=,=-===-.....O'-="'='====""="""'~'="'''''_'''=--===-~--='''"-''.==-''='''''''=-''''=--====--=-''''''_''''_-='''='='''''--==--==--==::.=.="',==

':-:~

';"::'-', ;;::...;_. _ _H.

_ ..;..-,;.--'

".

- __,_'_ _

,.~.

..,.. ::'-.._

_-••__

_";.~.::_

. CHAPTER 3. ORIENTATION

130

for all pairs [u, K] and [v, L], which are admissible for gluing, that is, which are endowed with matching ends u( +00) = v( -00) and K+ = L-. Let us denote the set of all such coherent orientations by CA' Moreover, we are interested in the group

r = {f E {+l,_l}A I f([u,K] #

-­

-~~.,,,...

-- __ ••.. __. __ .::~_...,"""~

" ''"'''::'''''''.

,_.",,...1":.-;.J~'

~-. .--;_~~.;:"""::,:;:-:;;:,e:;:;',;::;::,_.,;,,,"''''-"_'"''''''''-

,,~_:.':- '~.,._'.

with

,'''._,''''0'.'

'"

.:''''.'<'~''

"",~"",_",,,,,,,,,-. . _:~;..;.,,,,-

_ _ _ _ _ _-;..

-':;:..

" " ; : : : ' --.:":;=;~-:"~""""'_ ",::r.'

131

v- 1(t)=v(-t), AL-l(t)

=

Adt) ,

we consequently obtain unique orientations on the classes [v, L] E S+ already fixed by the condition

O'[v-l, L -1 ]#O'[V, L] = duo] . Obviously, this proves consistent in relation to the anchoring class

[uo] E

By the following proposition we verify that this definition makes sense. Proposition 3.16 Coherent orientations exist, i.e. CA acts freely and transitively on CA by -'>

-'C""

3.2. COHERENT ORIENTATION

[v,L]) = f([u,K])' f([v,L])}

with respect to pointwise multiplication.

r x CA

_... e_•••

i= 0,

and the group

O'[uo]#O'[uo] = O'[uo]

r

CA

(f,0') 1-4 f 0' (f, 0') ([u, K]) = f([u, K]) . O'[u, K]

By means of these two types of classes from S- and S+ anchored at Xo together with the associativity of gluing as far as the orientations are concerned, we are now able to accomplish the last step toward a coherent orientation. Considering an arbitrary [u, K] E A, we find unique classes

[vu,Ko,K-] E SProof In the first part of the proof, we present an algorithm for constructing such an orientation. It is crucially related to the associativity verified above (see also [F-H]). Let Xo EM be an arbitrary point in M and let A E End(TxoM) be non-degenerate and conjugated self-adjoint, for instance Xo E Crit f and A = H 2f(xo) for a Morse function f. Then, we consider Uo E CC'O(JR, M) to be the constant curve Uo == Xo = const

a at + A E Euo TM

vu(+oo) = u(-oo)

with

and

[wu,K+,Ko] E S+

with

wu(-oo) =u(+oo) .

Then, an orientation O'[u, K] is determined uniquely by the condition

O'[v u , K o, K-]#O'[u, K]#O'[w u , K+, K o] = O'[uo] , which is well-defined due to associativity. Moreover, associativity guarantees that this construction in fact provides a coherent orientation.

together with the operator

Ko = -

s- n s+

,

0

which is an isomorphism as A E COO (End(uoTM)) is constant. 6 Thus, we are able to fix the orientation 1 0 1* for the determinant space Det K o = lR 0lR*. Briefly, the class [uo, K o] E A, called the anchoring class, is oriented by [101*].

tions

0'1

We now analyse the r-action: Given two arbitrary coherent orienta­ and 0'2, we define

f : A -'> {±1} by

0'1

= f . 0'2

.

Obviously, this is equivalent to

The next step is to consider the subsets

and

S- = { [u, K] E A I u( -00) = Xo } S+ = {[u, K] E A I u(+oo) = xo}

0'2

6uQ = canst also implies the identity 'V t =

for

it·

.

Hence, given two classes [u, K] und [v, L] admitted for gluing, we are led to the identity

of A and to choose an arbitrary orientation du, K] for each class [u, K] i= [uo, K o] = [uo] from S-. Considering the one-to-one correspondence S- ~ S+ induced by the reversal of the time-parametrization, that is,

[v- 1,L- 1 ]ES-

= f . 0'1

0'1 [u, K]#O'l [v, L] = 0'1 [u#v, K #L] (f 0'2)[U#V, K #L] = fl . .. # ...]. 0'2[' .. # f([ # ])· (0'2[u,K]#0'2[V,L])

[v,L]ES+ =

f([

#

...]

J) (n u , K]O'du, K]#f[v, L]O'dv, L])

=='==-=----~~-

---.&...s====

132

This amounts to

z=~ ~-"-,:+;

7;;=(777::'-=

=~ ~-------ri-

-

~=.g.=~~~~~----;-;~;;~;,;..:;,-;"

~;'""'''''''"''-~'''''''''"'':''''''''-'''';[-'-''''':-~"''''''-'''''''=''''"'''''''''''''''-~~~~.~~'-:-'''"''-'-:- ..­

CHAPTER 3. ORIENTATION

1 = j[U#V, K #L] . j[U, K] j[V, L] .

Applied in both conclusion directions, this means that such a mapping j : A -+ {±1} represents an element of the group r, exactly if it transfers coherent ori­ entations again to coherent orientations. Hence we conclude that the asserted group action is in fact well-defined and transitive. Finally, it is straightforward to verify that this action is free. D

Chapter 4

Morse Homology Theory 4.1

The Main Theorems of Morse Homology

We introduce Morse homology theory by two main theorems about the exis­ tence of a canonical boundary operator associated to a given Morse function j and about the existence of canonical isomorphisms between each pair of such Morse complexes. These theorems appear as the essence from the theory on compactness, gluing and orientation which has been developed throughout the last three chapters. It is the isolated trajectories of the time-independent, time-dependent and the A-parametrized gradient flow which form the crucial features in the core of these main theorems. The following preparatory sec­ tion describes how we associate respective characteristic signs to these isolated traj ectories.

4.1.1

Canonical Orientations

Concerning the Fredholm classes [u, K] E A analysed in the last chapter, we henceforth denote for all trajectories u of the time-independent, time-dependent or A-parametrized gradient flow by [u] the class [u, Dul, where D u stands for the surjective l Fredholm operator yielded by the linearization of

a at + V' j,

etc.,

in local coordinates at u. Due to the surjectivity of these operators together with the representation

TuMx,y:= ker Du c H l ,2(u"TM) lexcept for the >..-parametrized gradient flow; see below

CHAPTER 4. MORSE HOMOLOGY THEORY

134

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

for tangent spaces of the trajectory manifolds, an orientation of the class [u] gives unique rise to a differential-topological orientation of the connected com­ ponent of Mx,y containing u. Now, in contrast with the notion of a coherent orientation, we find the following canonical orientations for the classes [u] E A in the cases of isolated trajectories u:

so that D 2 Ga.{3(>.., u>,) = DUA is onto, the natural isomorphism n(>.., u>,) : Det DUA

• In the case of the time-independent gradient flow, i.e. u E M~,y, ker D u turns out to be of dimension 1. We find a canonical orientation by -\1 f

0

u E ker D u

,

because these solutions of an autonomous differential equation are en­ dowed with a one-dimensional shifting-invariance. This orientation o([uJ) is briefly denoted by rUt].

n : Det D 2 Ga.{3 ~ A maxker DGa.{3 ,

01:,

I--'

is generally not a surjective operator. Restricting the analysis to a relative Morse index p(xa.) - p(Y{3) = -1, this means dimker DUA = 0 and dim coker D UA = 1 and therefore isolated >..-trajectories, we obtain by n the orientation 1 ~ (D 1Ga.{3. ,r on DetD 2 Ga.{3 = DetDuA from the canonical orientation 1 on A maxker DGa.{3 = IR. Here, we apply the canonical identification coker D 2 Ga.{3(>..,u>,) ~ R(D 1Ga.{3(>..,u>,)) We deal equivalently with the canonical orientations [> [>

1 ~ 1* on T(>"u A)M{;""YI3 and

1 ~ (D 1 Ga.{3(>.., u>,)·


on [u>" DuJ

M~o"YI3

T".(U) a[uJ = rUt]

2. for u E M~:~x/3:

T".(u)a[u] = [1 ® 1*]

3. for u E M X ",A'YI3:

4.1.2

T".(u)a[u] = [1 ® (D 1 Ga.{3(>..,u).

--+

[O,IJ ,

,)*J

The Morse Complex

Definition 4.2 Let f E COO(JR, M) be a fixed Morse function and Critk/={xECritf

I p(x)=k}

for O~k~n

the discrete subset of M containing the critical points of f with Morse index k. We define Ck(f) = Critkf ® IE as the free abelian group generated by the critical points with Morse index k, i.e. in particular Ck(f) = 0 for k < 0 or k > n. Given a fixed coherent orientation a

If we consider the I-dimensional components of M!!a" y13 , i.e. p(xa.) = p(Y{3), and regular >..-values, which means regular with respect to the projection map Jr :

1. for u E M{y :

H"'13

which was introduced on components of the parameter-trajectory spaces M!;",YI3' where

D 2 Ga.{3(>..,u>,) = D UA E 2: u>..•T M, U E M{;A y.o

H

T(>"u A)M x"',YI3

Definition 4.1 Given any fixed coherent orientation a as developed in the last chapter, we are able to associate characteristic signs T". (u) to the isolated trajectories by comparing the above canonical orientations for [u, D u ] with a, with respect to the three different types of isolated trajectories, that is

• As to the >..-parametrized gradient flow, we consider the natural bundle isomorphism from the last chapter (4.1)

~ ----4

identifies the canonical orientation 1 ~ 1* on [DuJ with that orientation on M~ ,Y/3 which is mapped onto the canonical orientation , of [0, 1] by the projection map Jr. Deciding whether two orientations [1 ® 1 *]u A1 and [1 ~ 1 *JU A2 of operator classes [DUA ; J at regular >"-values and within the same component of M~ ,Y/3 are equivalent amounts to analysing the pre-image of the projection map on the interval of >..-parameters; see also Figure 4.4 on page 146.

• As to the time-dependent gradient flow, i.e. for isolated trajectories in M~"'l3x , ker v'u vanishes, so that we may orient the trivial line bundle "', 13 Det [u] = JR ~ JR*. canonically by 1 ~ 1*.

01- u =

135

mapping (., .) : Critf x Critf

E

CA on M we now define the

--+

IE

by

(4.2)

(X,Y)C~{

L

T".(11),

for

p(x) - p(y) = 1 ,

~f

iiEM",y

0,

otherwise.

136

CHAPTER 4. MORSE HOMOLOGY THEORY

The pairing (x, y)U amounts to counting the connecting orbits with relative Morse index 1 between x and y equipped with their characteristic signs from the above definition. Here it is worth mentioning that the characteristic number (x, y)U corresponds exactly to the geometrical intersection number of

137

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

k ~ 1, straightforward computation according to the above definition leads to the equation

a2 x

L

=

L

(x, y)U(y, z)U Z

zECrih-2! yECritk_tf

L

WU(x) rf1 WS(y)/1R as described in the introduction, up to a suitable change of the noncanonical orientation of the respective critical points. This noncanonical ingredient is equivalent to the noncanonical choice of a coherent orientation. We shall il­ lustrate this interrelation between the present approach to Morse homology and the classical approach by Thom, Smale, Milnor and Witten in more details in Appendix B. Since Critk/ is the generating set of the group Ck(f) for k 0, ... ,n, we extend (', .)U to

(4.3)

( " . )U : C.(f) ® C.(f)

~ ;£

a;;x =

~

L

Ck-1(f)

(x, y)U y

(4.4)

Theorem 7 Given any Morse function f and any coherent orientation a, the family of homomorphisms

(Ck(f),

ankEZ

=

a2x =

L

L

a;;_l 0 a;; -

0 for all k = 0, ... ,n

Within this monograph, we call this chain complex the Morse complex , in contrast to classical theory, which denotes by Morse complex the associated cellular complex2 . Proof. Given any x E Ck(f), Le. without loss of generality x E Critk/ and 2associated to

f by

homotopical equivalence with M, see [M1]

L

Tu(U) . Tu(V)

Z

Since we already know that each of these equivalence classes contains exactly two different broken trajectories, we could deduce the assertion in the special case of ;£2 coefficients from the identity

L

1 == Omod2 .

(u.v)E[(u.v)]

The analogous general step concluding the proof,

L

(4.5)

Tu(U) . Tu(V) = 0

(u,v)E[(u,v))

will be proven in the following lemma by means of the interrelations between gluing, coherent and canonical orientation, Le. briefly by oriented cobordism theory for simply broken trajectories. Thus, the theorem is concluded from

(4.5).

0

Lemma 4.3 Let x, y, y',

(C.(f), aU)

represents a chain complex, i. e. it holds that

Z

ZECrih_2! [(u.v)]EM""./~(u.v)E[(u,v)]

yECritk_l!

This definition leads us to the first main theorem within this work on Morse homology:

Tu(U) . Tu(V)

where we denote by M x •z the set of simply broken trajectories between x and z, as was defined in the gluing Section 2.5. Now, resorting to the cobordism equivalence, we may reorder the double sum on the right hand side, obtaining

and we define a homomorphism sequence (ak)k=O .... ,n associated to (Ck)k=O, ...•n by

a;; : Ck(f)

L

zEC rit k_2! (u,v)EM"".

Z

E Crit

f be critical points satisfying

J.L(x) - 1 = J.L(Y) and let

= J.L(Y') = J.L(z) + 1

~!~!

,

~!~!

(U1, Vl) E M x•y x My,z' (U2' V2) E MX.yl x My',z be equivalent broken trajectories, i. e. (U1' VI)

rv

(U2, V2)

within M~.z .

Then, with respect to any arbitrary coherent orientation a on M, the identity

Tu (U1) . Tu (V1) holds.

=

-Tu (U2)' Tu (V2)

138

CHAPTER 4. MORSE HOMOLOGY THEORY

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

139

t·I··.··· ." I,:

Proof· The cobordism-equivalence signifies per definition that both broken tra­ jectories are mapped by the respective gluing ope~tion # into the same com­ ponent of the unparametrized trajectory space Mx,z, which is diffeomorphic to (-1,1). The strategy of proving this lemma is to compare the orientations which are induced by the gluing operation on this component from the re­ spective canonical orientations. The following relation between the canonical orientations defined above has to be verified:

[Ult] # [Vlt]

(4.6)

=

-[U2t] # [V2t] .

irl~t It,.; i ';

,.i~"-

I ,:>~Zt:-

denote the diffeomorphism which endows the one-9imensional connected com­ ponent with a fixed orientation vector, e E T1f;(po)M x,z 0, i.e.

D¢· e =

11~11.

a[Ul #Vl]

(Ul,V!l.::::(U2,V2)

a[u2#v2]

l1EClI.

Mx,y xffi. (u, T)

(4.7)

-..::...

Mx,y

u

f-+

T

in relation to both versions of the trajectory gluing, for parametrized as well as for associated unparametrized trajectories. Let (u, v) be a fixed unparametrized broken trajectory from the connected component Mx,zo, Po an appropriate gluing parameter and E > 0 fixed, so that we obtain the following commutative diagram, {U}c-€,)€

x {v}(_€,€)

#po • Mx,z

°

J/R

~1 (po-E,Po+E)

{U}c-€,)€

x {v}c-€,€)

((~, ~), (~, -~)) , where the embedding rp identifies the pair of tangent vectors

(~, -~) exactly with :p E Tpo (Po -

E,

E

T(uo,vo) (Mx,y x My,z)

Po

+ E).

Referring to these fixed orientations, we now have to determine whether the embedding #po acts in a preserving or in a reversing manner. According to the above commutative diagram, it identifies the pair of tangent· vectors (~, -~) with the pre-image of the vector D'lj;· :p with regard to the differential of the projection map It likewise identifies the pair (~, ~) with the vector -aa E T # vM z ° canonically induced by time-shifting. Hence, u x Ta PO , the question of orientation preserving or reversing by the gluing operation #po is reduced to the comparison of the fixed vector with the tangent vector on (-1,1) which is induced by ¢ 0 'lj; together with :p E Tpo (Po, -E, Po + E). Ex­ actly at this stage we derive different characteristic signs for the two equivalent broken trajectories (Ul' VI) and (U2' V2).

-fffi..

:s

Assuming the convergence lim ¢(Ul#pVl) = 1

P~OCJ

.. Mx,zo

'lj;

(-1, 1)

by

a[u2] # a[v2]

Hence, we have to analyse how the canonical orientations behave with respect to simply broken trajectories in relation to the gluing of trajectories. We consider the diffeomorphism from the time-shifting action

so

As a consequence, time-shifting (4.7) and the orientation [e] on Mx,z ° Mx,zo /ffi. induce a fixed orientation (a~u' e) on Mx,zo. On the other hand, we may also consider the canonical orientations on

Then, according to the definition of T l1 (Ut}, etc., the assertion is concluded from the coherence condition and the cobordism-equivalence:

a[Ul] # a[vI]

asa E T

and respectively

together with the embeddings

lim ¢(U2#pV2)

=

-1

P~OCJ

rp

: P f-+ (U(p-po), V(po-p))

'lj;

: P f-+ u# p v

~

and

.

Note that the appropriate choice of Po depends on

¢: Mx,zo ~ (-1,1)

f.

Now, let

for a given fixed diffeomorphism ¢, we observe that

• :s • -:s

by 'lj;l (-)

= Ul i VI and ¢ and with

by 'lj;2(-)

=

U2#.V2 and ¢ .

:p is identified with

CHAPTER 4. MORSE HOMOLOGY THEORY

140

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

by

x

•

(4.8)

(x., xp

}~p ~ {

~

Ta(U), haiJ

for J1.(Xo.)

=

J1.(X{3) ,

uE M 0'J:a..:r.13

0,

otherwise

for the generators (Xo., X(3) E Crit r x Crit f{3. Once again, the compact­ ness result for the trajectory space M~:~xiJ consisting of isolated trajectories guarantees the finiteness of this sum. Consequently, we define the group homo­ morphisms by analogy with the above 8 k , that is

.y'

y.

141

epk(ho.{3) = ep~o. : Ck(r) _ Ck(f{3)

L

ep~o.xo. = (Xo., x{3 }~{3X{3 xiJECritkfiJ Now, the fact that the latter sum is finite is derived from the compactness results for Morse homotopies as was stated in Corollary 2.46. Note that this relies essentially on the coerciveness of the considered Morse functions.

•

z

Figure 4.1: Cobordism-equivalence for 8-trajectories Thus, we deduce the relation asserted in (4.6) between the orientations in­ duced by gluing of equivalent broken trajectories. This adverse orientation induction is illustrated in Figure 4.1. The arrows on the broken trajectories mark the canonical orientations ([Ut], -[VtJ), which under the gluing opera­ tion correspond to the respective orientation of the one-dimensional connected component 0 according to the respectively indicated weak convergence of unparametrized trajectories. D

Mx,z

4.1.3

Proposition 4.5 The family of morphisms

ep~o. = (ep~o. : Ck(r) -

Ck(l{3)) k=O, ... ,n

forms a chain homomorphism ep~o.

: C.(lo. _ C.(I{3) ,

i. e. it holds

The Canonical Isomorphism

Given two Morse functions, we now intend to derive a chain homomorphism between the associated chain complexes C.(lo.) and C.(I{3), which gives rise to a canonical isomorphism between the respective homology groups. Actually, we will obtain this chain homomorphism in the same way as we extracted the canonical boundary operator from the negative gradient flow of a fixed Morse function. As to the isomorphism, 'canonical' means the independence from any further assumptions apart from rand f{3.

Definition 4.4 Let rand f{3 be any Morse functions on the manifold M, let ho.{3 be an arbitrarily associated Morse homotopy and let us choose any coherent orientation a. By strict analogy with Definition 4.2, we define

(', . }~{3: (Ck(r))k=O, ... ,n x (Ck(l{3))k=o, ... ,n

The first step toward the second main theorem of Morse homology consists of the verification of the fundamental functorial relation for chain complex morphisms:

-

Z

8f(f{3) 0 ep~o.

=

ep~~l o8nr) for all k = 0, ... , n .

Proof. The proof follows that of Theorem 7. Using the short notations

80.

= 8nr),

8{3 = 8f(l{3)

we compute

(8{3 0 ep{3o. - ep{3o. 0 80.) (Xo.)

L

L

L

L

8{3( Ta(Uo.{3)X{3) - ep{3o.( Ta(Uo.)Yo.) /L(XiJ)=k UaiJEM"'a''''iJ /L(Ya)=k-l uaEM"'a,Ya

L

n(xo., Y(3) Y{3 /L(YiJ)=k-l

CHAPTER 4. MORSE HOMOLOGY THEORY

142

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

143

xi"

where

=

n(x"" Y(3)

L

L

L

L

L

L

Ta (U",{3) . Ta (U{3)

M"a{3 ~ M~ f{3 ( ) k J1, x{3 = u",{3E "''''''''{3 u{3E "'{3'Y{3

Ta(U",)· Ta (V",{3)

~f'" M"",{3 k 1 J1,(Y", ) = - u",EMx""y", v",{3E Y""Y{3

eX{3I

X{3e

We now prove the identity

n(x"" Y(3) = O.

(4.9)

in a way that is comparable with Lemma 4.3. The analysis of the cobordism­ equivalence for mixed broken trajectories with fixed endpoints

(x"" Y(3)

E

U{3

Critkr' x Crih_d{3 ,

e

this time requires a separate discussion of two different cases related to the order of the trajectories: (a)

( u",{3,

.)

u{3

E

(b) (U"" V",(3) E

Mha.{3 ~f{3

X""X{3 X M X {3,Y{3

Nt!'"

xooY~

xM

and

h ",{3

YcoY/3'

Starting from the broken trajectory (u",{3, u(3), we observe two different possi­ bilities for cobordism-equivalence with regard to this distinction of cases:

(a) (U"'{3, u(3) ,...., (w",{3,v{3) E Mx""x~ x M x #,Y{3 for a x~ E Critkf{3. Analysing the orientation induction by the gluing operation for mixed broken trajectories, we are led to an identification of the canonical orien­ tation ( [1 0 1*], [U{3,t]) on the component of (v",{3, u(3) in M x ""x{3 x M X {3,Y{3 with that orienta­ tion on the one-dimensional component M x ""Y{3°, which corresponds to a decreasing parameter p regarding the gluing of broken trajectories. Ac­ tually, we observe this orientation correspondence at both ends, because we deal both times with broken trajectories of type (a) according to the above specification (see Figure 4.2). Hence, this case leads to the identity

(4.10) (b)

Ta (V",{3)' Ta(U{3)

= -Ta (W",{3) . Ta (V{3) .

(U",{3, u(3) ,...., (u"" v",(3) E Ntx""y", x M y ""Y{3 for a Y'" E Critk-lf"'· In the case of an equivalence with a mixed broken trajectory of type (b), this trajectory is endowed with the canonical orientation ([U""t], [1 0 1*]) ,

Y{3 Figure 4.2: Cobordism-equivalence for mixed broken trajectories of type (a)-(a) which now corresponds to an increasing gluing parameter (see Figure 4.3). Thus, the gluing operation induces at the cobordant broken trajectories of different type the same orientation on the associated one-dimensional component M x ", ,Y{3 o. Hence, we verify the relation (4.11 )

Ta (U",{3) . Ta UL(3)

= Ta(U",) . Ta (V",{3)

in this case. As a consequence, we obtain products of characteristic signs with different signs in the case of mixed broken trajectories of the same type as in (4.10) and with equal signs in the case of different types as in (4.11). Since the cobordism­ equivalence from the compactness-gluing-complementarity yields a one-to-one correspondence between mixed broken trajectories regardless of type, the above case distinction accomplishes the proof of the asserted identity

n(x", , Y(3)

=

0

and thus of the proposition, too.

0

The next step now consists of verifying that the induced homomor­ phisms on the level of the homology groups, tP~'" : H* (j"') .-. H* (j{3)

are in fact independent of the choice of the actual Morse homotopy j'"

h:;: f{3.

CHAPTER 4. MORSE HOMOLOGY THEORY

144

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

145

r

associated to the fixed endpoints (x Ol , YfJ) E Crit x Crit ffJ. Here, we use the brief notation M x", ,Y{3 (A) for the trajectory space associated to the fixed parameter A. 4 The ascending of the degree of the operator

XI'

W~Ol :

C. (r)

---+

C.+l (ffJ)

is due to the fact that, within this framework, we obtain isolated trajectories exactly if the relative Morse index is .yOl

XfJ·

{L(X Ol ) - {L(YfJ) = -1

Under this condition the finiteness of the set M~~:{3 is provided by the re­ spective compactness result. Again, these isolated trajectories can be counted with characteristic signs and we define UfJ Wk : C k (r)

• YfJ

T". (u>.)

Thus, we have to compute the difference of the terms a fJ

0

W~Ol(XOl)

=

L

L

T".(U>.) afJzfJ /L(z{3)=k+l (>. , u>.)EMH",{3 XCt,Z~

-

W~~1

0

af

L [L

(4.13)

In other words, w~fJ represents a chain homotopy between the chain complexes C.(r) and C.(ffJ).

W~~1

E

[0,1]

X

1,2

'PX""Y{3

3We extend the finite family C.(f"') by Cj(f"')

0

L

aOl(x Ol ) -

L

T".(U Ol ) WfJOlyOl

/L(y",)=k-1 u",EM~:,y",

L [L

L

L

T".(UOl ) . T".(V>.)] xfJ /L(x,B)=k /L(y",)=k-1 u EMJ'" (>' v>.)EMH"',B XQ,YOt. I YCt1Zfj Q

with the homotopy parameter A, so that, due to the regularity result in Theorem 2, we obtain the finite-dimensional manifold

(A, u)

L

and

(4.14)

H",{3 OlfJ h OlfJ o -~ h 1

L

T".(U>.). T".(UfJ)] xfJ /L(x{3)=k /L(z,B)=k+l (>. IU>.)EMH",{3 U EM~ J,B Xa:'Z/3 13 Z{3IXfj

Proof. At this stage we finally have to apply the results about transversality, compactness and orientation as far as the spaces of A-parametrized trajectories are concerned. Let us choose a regular homotopy

_ {

L

( >. , U>.)EMH",{3 XO:'%/3

Ck(ffJ), k E Z

~~,~ - ~g,~ = ae+l 0 W~Ol

H",{3

=

( XOl , zfJ)

satisfying the identities

M x""Y{3 -

(xOl , zfJ) zfJ /L(z{3)=k+l

~gOl and ~~Ol.

fOl and ffJ together with their associated chain morphisms Then there is a famil71 of morphisms

(4.12)

L

,T.OlfJ '¥k XOl

Proposition 4.6 Let h~fJ and hrfJ be two regular Morse homotopies between

---+

Ck+l(ffJ)

for k E Z on the generators XOl E Crit fOl by

Figure 4.3: Cobordism-equivalence for mixed broken trajectories of type (a)-(b)

W~Ol : Ck(r)

---+

I u E M H",{3} x""Y{3 (A)

=0

for j

< 0 or

j

> n.

with regard to the right hand side of equation (4.12) and (4.15)

(~~Ol

-

~gOl)

(xOl ) =

L

[L

T".(U1) -

/L(X,B)=/L(x"')Ut EM~; ..,,B 4This is not necessarily a manifold!

L

T".(UO)] xfJ

uoEM~~ •.,,B

~~:ry]~~~~~~~~~~...:a~,;!,o-~~~=~~~~~~-~_.,,",,,,,,,,~====,,,.~-,,,,,.=-=~==~---_._._--------

146

CHAPTER 4. MORSE HOMOLOGY THEORY

with respect to the left hand side. In principle, all steps which have to be gone through are known from the above propositions as applications of cobordism­ equivalence for trajectory spaces. In this framework, however, we have to apply our results on compactness, gluing and orientation to the 0- and I-dimensional manifolds of the A-parametrized trajectories. This leads to a slightly richer vari­ ety of possible cases to consider. This time, the connected components in ques­ tion may also be manifolds with boundary. Hence, regarding the cobordism­ equivalence, we have to take into consideration more separated cases. The sketch in Figure 4.4 suggests a geometrical visualization of the closed subman­ ifold M;L,z~ C [0,1] X p~~2,z~ for J.l(x a ) = J.l(zj3). There, the ends of the one-dimensional components, which have a boundary in the weak sense of con­ vergence toward mixed broken trajectories, are indicated by open brackets ")'.

THE MAIN THEOREMS OF MORSE HOMOLOGY

4.1.

147

Obviously, case (b) does not play a role in our cobordism-equivalence, be­ cause those closed components neither contain boundary points in the strong sense, that is, curves u E MXQ'x~(O) UMxQ,x~(I) as trajectories for the time­ dependent gradient flows yielding M~~,x~ and M~~,x~, nor boundary points in the weak sense, that is, mixed broken trajectories, composed of trajectories v>. E MXI,ZI(A) satisfying J.l(x') - J.l(z') = -1 and respective a-trajectories. Concerning the oriented cobordism-equivalence for trajectory spaces, the gluing operation associates to each mixed broken trajectory

(u>., uj3) E Mx""z~(A) x M!:,x~

(u a , v>.) E Me,y", x My""x~(A)

and

building up the terms in (4.13) and (4.14), a one-dimensional connected compo­ nent of M;;Q'x~ of type (c) or (d). Additionally, regarding the terms in (4.15), the trajectories u a j3 E Mx""x~ (0) U MXQ'x~ (1) lead us to connected compo­ nents with boundary in the strong sense, that is, of type (a) or (d). Hence, the discussion of the cobordism relations in the proof of Proposition 4.5 has to be extended by the cases (a) and (d) of manifolds with boundary. As to (a): The connected component M:",x~ 0 in question is diffeomorphic to the compact interval. As we analysed in the first section of this chapter, the natural isomorphism n induces the following orientations of the component y 0 from the canonical orientations [1 @ 1*] of the classes lUi], Ui E "" ~ M~:y, i = 0,1, at the boundaries of the interval, A = 0,1:

M;;

• A =

°"" "

orientation toward the interior of the component,

• A = 1 """" outward directed orientation. Given the boundary curves

aM;; Xa 0 = {uaf.l, Vaf.l} C M~o Xa UM~l Xa , fJ fJ O:'IJ

O:lIJ

OO/J

we consequently have to distinguish the following two cases: Figure 4.4: The I-dimensional manifold with boundary of A-parametrized tra­ jectories With regard to the components of the one-dimensional A-parame­ trized trajectory spaces, we have to consider separately the following diffeo­ morphism types: (a) [0,1], with aM;L,xiJ C Mx""xiJ(O) u M xQ ,xiJ(I),

(b) SI, (c) (-00,00), (d) [0,00) and (-00,1].

1. Assume {u a j3, v a j3} C M~~,xiJ or M~~,xiJ' Then, in both cases, the

canonical orientations of u a j3 and va j3 give rise to opposite orientations on the one-dimensional connected component, such that the identity

er[u a j3J

=

er[va l3]

implies the relation T(T(U a j3)

= -T(T(Va j3)

2. Considering the situation ua j3 E M(A = 0), va l3 E M(A = 1), we obtain at both ends the same orientation on the component from the canonical orientations of the boundary points, so that we are led to the identity T(T(U a j3)

= T(T(V a l3)

.

CHAPTER 4. MORSE HOMOLOGY THEORY

148

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

Summing up these results, we may reduce the difference

149

Xi'

(ipfo - ipgo) (X o ) u~

from (4.15) to trajectories u o{3 E M x",xl3(>' = 0,1), which solely bound com­ ponents M~,xl3 0 of type (d).

As to (c): Considering connected components M~,xl3 0 of the diffeomorphic type (-00, (0), we no)\' have to deal with weak convergence toward mixed broken trajectories at both ends. The natural isomorphisms n(A, u>.) : Det D u),

,

Z{3.

.z{3

---=-. AmaxT(>.,u),)M~

together with Proposition 3.14 enable us to reduce this case to the analysis in the proof of Proposition 4.5. Once again, we distinguish two types of cobordism­ equivalence by the order in the mixed broken trajectories:

•

X{3

• (u>., U(3)

E

M X",ZI3(A) x M{;,xl3 as in the right hand side of (4.13) and

• (u o , v>.)

E

M~:,y" x M y",XI3(A) as in (4.14).

Figure 4.6: Induction of orientation in opposite directions

The associated case distinction for the equivalence of broken trajectories is illustrated within Figures 4.5 and 4.6. We observe that the difference of both

mixed broken trajectories, which the gluing operation associates to the one­ dimensional components M~,xl3 0 of type (d). Thus, it remains to accomplish the analysis of the cobordism-equi­ valence for components of type (d), which, due to the discussions up to this point, establish the decisive link between the left hand side and the right hand side of the chain homotopy equation (4.12).

Xi'

~Uo,tJ ·Yo

Z{3.

v>. U{3

• x{3

As to (d): On the one hand, as we know from the discussion of case (a), the canonical orientation [1 0 l*J induces an inward pointing orientation on the entire component 0 [ M H"13 x",xl3 ~ 0, (0) at boundary trajectories Uo E M X",XI3(A = 0) and an outward pointing orien­ tation at boundary trajectories Ul E M X",XI3(A = 1). On the other hand, the case distinction for the orientation induction for mixed broken A-trajectories referring to the canonical orientations and the natural isomorphism n(\ u>.) yields an inward pointing orientation for the order

(u>., u(3)

E

M X",ZI3(A)

X

-fl3 M z13 ,xl3

Figure 4.5: Equally directed induction of orientation

and an outward pointing orientation for the mixed broken trajectory with the order -f" (Uo , v>.) E Mx",y" x M y",XI3(A) .

right hand sides (4.13) and (4.14) may also be reduced to the computation for

An illustration is given by the sketches in Figure 4.7. Hence, in relation with

,..-

-------_._-----,

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

CHAPTER 4.. MORSE HOMOLOGY THEORY

150 x'"

x'"

•

•

II • Zp

Uo

• y",

Uo

, etc.

151

Summing up this last analysis, we have obtained the result on the canonical homomorphism ~'" : H': (f"') - t H': UP) , associated to given Morse functions f'" and fP together with a fixed coherent orientation, by additionally taking into account the A-parametrized trajectory spaces. The next proposition establishes the final independence of the homology groups from the choice of Morse function .

rr

Proposition 4.7 Each ordered triple of Morse functions f"', fP and to­ gether with the associated homomorphisms ~' fulfills the composition rule

Up

JP

• xp

• xp

the boundary trajectories Ul E

M

x ",xl3(>\ =

0,1) ,

we distinguish the following four combinations:

(4.16)

•

T,,(U>.)· T,,(Up) = -T,,(Uo),

•

T"(U,,,)' T,,(V>.) = T,,(UO),

•

T,,(U>.)· T,,(Up) = T,,(ud and

•

T"(U,,,)· T,,(V>.)

=

L

/L(y,,)=k-l

L U1

L

C

=

(~(hP1)

0

~(h"'P)) . C

',..,..

L

'13'%13

L

i=l

so that relation (4.16) is satisfied. We consequently deduce the homological identity 0

~"') ({cd)

from Proposition 4.6 for each class

T"(U,,,) . T,,(V>.)

{Ck} E Hk(f"'), k E Z

u"EM~~...... ' ya: (>.,VA)EMyH_ 0:, .... /3

L

L aix~,i E Ck(f"'), k E Z ,

J"'({cd) = {~"'(Ck)' Ck} = {(~P o~"') (Ck)} = (JP

T,,(U>.)· T,,(Up)

(>.,uA)EM~" %Q UQEMf13

EM"" '%13 (1)

H': (f1) .

n

L

T,,(Ul)

~(h"'1).

Ck =

We are finally able to deduce the asserted identity for (x"" xp) E Critki'" x CrihfP:

/L(zl3)=k+l

-t

is satisfied for all chains C E C*(f"'). Fortunately, this is not necessary with regard to the required result for the homology groups. It is sufficient to find a homotopy hQ;1 (R(Ck)) according to the construction of the gluing for each fixed chain

-T,,(U1) .

L

~'" = J'" : H': (f"')

Proof. At this stage we take advantage of the freedom of choosing an appro­ priate Morse homotopy, established by Proposition 4.6. Considering the homo­ morphisms JP and ~'" induced by ~(hP1) and ~(h"'P), respectively, we now use the gluing construction for simply broken trajectories consisting exclu­ sively of homotopy trajectories associated to hP1 and h"'P. Due to Proposition 4.6, every Morse homotopy h"'1(R) constructed in this way must yield the same homomorphism J"'. But in order to be able to apply the gluing result from corollary 2.61, we are only allowed to put a finite number of critical points x"" xp, x 1 into relation with each other by isolated h-trajectories. Hence, we generally cannot find a homotopy hQ;1 such that the identity

Figure 4.7: Cobordism-equivalence of type (d)

uo,

0

T,,(UO)

uoEM"" '''13 (0)

o

Thus, without loss of generality, we may start from a fixed Ck = x", E Critkf"', so that there is only a finite number of pairs (xp, x 1 ) E CritkfP x Critkrr with M h "l3 X M h13 "1 -J- 0 Xo.,X/3

Xj3,x')

r

----~

152

CHAPTER 4. MORSE HOMOLOGY THEORY

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

We set the constant

153

Definition 4.8 Finally, this functorial concept leads us to the independence of R = maxR(x a , xf3, x')')

our homology theory from the choice of a concrete Morse function. Considering the product groups

according to Corollary 2.61. Thus, the identity

LL

L

Hk =

Tu (U a f3) . Tu (Uf3')') x')'

x", xfj U"fjEMx",xfj Ufj",EMxfj,x..,

(4.17)

LL

L

H k = { ( ... , {en, ,." {d~}, ... )

together with the bijection x

#n·

Mh"fj

Xcoxp

Xp,X'"'(

E

Hf

I {d~} = cP~a . {ck}, ... }

which is well-defined due to Theorem 8. This homology group, independent of the Morse functions, actually describes the same as the inverse limit ~ H k

~ Mh",fj

xo:,x,..

(r) with resped to the isomorphisms cP~a. This concept will be called the

from Corollary 2.61, proves the assertion. Here, the identity Tu (U a f3)' Tu (Uf3')')

Hf(r), k E IE ,

we define H k as the subgroup

Tu (U a f3#R Uf3')')

.x", xfj U"fjEM"",xfj Ufj..,EMxfj,x..,

. Mh"fj

II

f'" Morsefct.

identification process.

= Tu (U a f3#R Uf3')') = Tu(U a')')

follows from the relation between gluing and induced orientation as already analysed above. D

4.1.4

Let us now sum up the results from the last three propositions in the second main theorem of Morse homology:

Throughout the rest of this chapter we assume without loss of generality that M is a connected manifold. Since we intend to relieve our Morse homology concept from as many noncanonical assumptions and inputs as possible, the last item we have to get rid of is the coherent orientation 0". Therefore, we have to analyse how the homology groups H~; change if we replace 0"1 by 0"2. In a way comparable to the case of Morse homotopies, we will look for canonical isomorphisms, which enable us to identify uniquely the elements of the groups sequence

Theorem 8 Given a fixed coherent orientation

0" on the manifold M and two arbitrary Morse functions fa and ff3, there is a canonical isomorphism between the associated families of homology groups

<'-If.:

}

cP~a : H~(r) ~ H';(ff3) , and the following functorial relations are fulfilled for the family {cP~a ff3 Morse functions}:

• cPI •

f3

cP~a

0

Ir,

~;

{H~ }{u coho orientation}

,~1:

as in the inverse limit process. Actually, the key for this problem lies within the group action described in Proposition 3.16,

cP~a = cPI a

= idH~(J"')

Topology and Coherent Orientation

r

.

Proof First, the isomorphism property is a consequence of these functorial

relations as we see from ;F,.af3 ;F,.f3a

'¥.

0

'¥.

'd

= 1

H~(J"').

and vice versa. The fact that cP~a actually reproduces the identity is concluded immediately from the constant Morse homotopy

-+

CA .

Our aim is to extract homology isomorphisms from these transformations r.p E r of coherent orientations. For the sake of simplicity we first illustrate the procedure with a fixed Morse function f and consider the associated homology groups H':,(f). Obviously, we may put all coherent orientations 0" E CA coinciding on the iso­ lated trajectories of the negative gradient flow of f together in one equivalence class, that is

r{;r. D

x CA

(4.18)

{ 0"0 } = { 0"

E CA

= ([u, DuD I for all DuD E Mt,y, x, y E Crit f O"([U,

U

0"0

}

.

.

~ -~

-

~

-.

-""""

--

154

-

--- --~--

u-

--~

--- --­ - -- -~~

CHAPTER 4. MORSE HOMOLOGY THEORY

Hence we define the set of equivalence classes as CA consider the group of transformations

f- - )!":t:'

together with the action

G(f) (4.19)

X

(cp, (1)

-

CA

f---*

----t

-

CA,

_0-

---

"

_0

--'"'"

•

_n_

_E"" ----

_~~O~"

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

= CA/",' Let us now

G(f) = {±1}Crit f

~-

Ll:.~.c

--~- --- -~---&

155

whose supporting curves may be connected to an n-dimensional 'Mobius band' (u . V-I )*TM. We remember that we might put the original definition of the equivalence of such Fredholm operators along curves in a more general form, such that operators fulfilling the condition WI ((u.v- I )*TM) = 0 are admitted for simultaneous orientation. Already this weaker condition, which admits more possible coherent orientations, implies the fundamental identity aU 0 aU = 0 of a boundary operator. Example Let us consider the non-orientable manifold JlD2(IR) endowed with a suitable Morse function as indicated in Figure 4.8. Then, it is due to the chain

cpa,

(cpa)[ux,y] = cp(x)cp(y)a[ux,y] for

[UX,y] = [ux,y, Du""y], Ux,y

E

M!,y

ZO

•

In fact, G(f)/± forms a subgroup of r. The next observation is that, given any such orientation transformation cp, we can indicate uniquely a suitable isomorphism of the associated homology groups H';l (f) and H';z (f): Choosing cI>\O : (0*, aU)

(4.20)

----t

V2

VI

(0*, a\OU)

cI>\O • x = cp(x) . x, x E Crit f

,

•UI

,

we compute

·----:..... x

2

U2

---­

<:U(ux,y) = cp(x) cp(y) Tg(Ux,y), aUx

=

L( x, y )U y = cp(x) L( x, y )\017 cp(y) . Y y

y

This leads to the identity cI>\O aU x

V2 cp2(X) L(X, Y )'fJucp(y). Y

•zO

y

(4.21)

VI

L(x, y)\OucI>\O. Y y

a\OU cI>\O y . Thus, cI>\O represents a chain homomorphism, which is one-to-one due to the involutivity cp2 = id. Consequently, each transformation cp E G(f) gives rise to a homology isomorphism. However, there is the problem that the endpoints of the [u, K] E CA merely determine the transformations 'P up to the sign ±cp. This amounts to stating that the associated isomorphisms of homology are only determined up to the sign of their degree of modulus L Fortunately, by introducing an additional parameter, we are able to remedy this flaw within our orientation concept. At this stage we shall discuss in a more particular way the coher­ ent orientation of equivalent operators, which coincide at their endpoints, but

Figure 4.8: a-trajectories on JlD2(IR)

a

complex property 2 = 0 for any arbitrary coherent orientation with respect to the weaker equivalence relation just proposed above that we compute the following relations for the characteristic signs of isolated trajectories:

(4.22)

Tu(ud . Tu (V2) Tu(ud . Tu(vd

-TU(U2) . Tu(VI) -Tu (U2) . Tu (V2)

These equations do not, however, determine the characteristic signs Tu(Ui)' Tu(Vi) uniquely. Since UI • u 2 I and VI . ViI represent Mobius bands, the

CHAPTER 4. MORSE HOMOLOGY THEORY

156

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY This yields a generalization of {± 1 }Crit f in the sense of

operators

(Ul' D u ,) 'f (VI, D v ,) 'f

Crit f

(U2, D U2 ) (V2, D V2 )

x

cannot be equivalent according to the weaker version despite the coincidence of the endpoints. In fact, we may suggest exactly two different coherent ori­ entations up to some transformation from G(f), which yield non-isomorphic homologies, namely:

(a)

(Hg', Hf', H~') ~ (2, 2 2 , 0)

(b)

(Hg2, Hf2, H~2) ~ (22, 0, 2)

(22 ,0,2) ~ (H 2 ,Hl,HO) This means that, already at this stage, we find a hint of Poincare duality, which is immanent to Morse homology. It is closely connected to the choice between positive or negative gradient flow for the definition of boundary operator. As to the example, in order to comply with the dimension axiom, we have to decide on case (a). As we shall see in Lemma 4.13, this is related to the condition of the equivalence relation for operators in Definition 3.7. That is the condition that we simply demand the coincidence of the trivializations at the negative end, provided that we start from the Morse homology associated to negative gradient flow. Otherwise, choosing the positive end for the coincidence condition would lead us to the cohomology turned around by reflection. Definition 4.9 At first we define an appropriate generalization of G(f) with respect to arbitrary Morse functions. Hence, we have to consider the set of Hessians

£ = {A

E d(TM)

mEn

I

Am: TmM ----+ TmM } non-deg. and conj. self-adjoint

~

= {±1}£

instead of Crit f. Let be a group with respect to pointwise multiplication together with the action

(rpO") [u x.y, K;, K;;]

=

rp(K;)rp(K;;)O"[ux,y, K;, K;;]

£

~ H 2 f(x)

.

which additionally are compatible with the isomorphisms

~~a : H~(r) ~ H~(f{3) on the level of the homology groups. Hence, the problem indicated above may be expressed as follows: ~/± = r ; that is, there are exactly two transformations rp E ~ associated to a given pair (0"1,0"2) of coherent orientations, such that the identity 0"2 = rpO"l is satisfied, namely ±rp. In order to attain a unique identification of the homology groups by means of the isomorphisms rp* = ~'P associated to the rp E ~, we crucially require the functorial behaviour (4.25)

(rp' . rp)*

=

rp: 0 rp*

This condition prevents us from choosing erratically between +rp and -rp. Actually, the problem of a natural selection process with respect to functo­ rial behaviour lies in the relative constitution of a coherent orientation and its transformations. These transformations are given by rp(x)rp(y) with respect to the pairs (x, y) of endpoints. Thus, the relative character of these transfor­ mations is in contrast to the absolute character of a mapping rp : £ ----+ {± 1}. In order to fix such a mapping rp uniquely, we need a characterization of rp by Fredholm operators, which are endowed with only one end within £. This means that, instead of merely orienting the relative, connecting trajectories from Mx,y = WU(x) rt1 W8(y), we have to find an absolute fixing, for example by means of an orientation of some stable manifold W 8 (y). Thus we shall develop the following concept of the so-called 'one-sided' operators to find a solution for this problem. Let us first regard the trivial case, M = R.n. Let A + E S be a conjugated self-adjoint operator on R.n and A E Coo ([0,00], End (n, R.)) endowed with A( +00) = A +. In this situation, we consider [0, 00] c R to bear the differentiable structure as a submanifold with boundary within iR. We consequently define the one-sided operator

~

x CA ----+ CA (rp, 0") ~ rpO"

'----+

Once again by analogy with (4.20) and (4.21) the rp E ~ give rise to isomor­ phisms (4.24) ~'P : (C*(f), 80') ~ (C*(f), 8'P0') ,

and

If we make use of the original equivalence relation, however, which orients the 'Mobius bands' in a unique way, exactly one of these two possibilities is selected. Referring to this example, we observe that possibility (b) violates the known identity for the homology of connected manifolds, Ho(M) ~ 2, which is principally related to the dimension axiom. Actually, case (b) yields nothing more than the classical cohomology of lP'2(R.) turned around by reflec­ tion, that is

(4.23)

157

(4.26)

SA: X = H l ,2([0,00),R.n)

----+

y = L 2([O,00),R.n)

u

~

it+A·u.

_______

158

CHAPTER 4. MORSE HOMOLOGY THEORY

Referring to the methods which were developed throughout the Fredholm Sec­ tion 2.2.1, it is straightforward to show that SA represents a Fredholm operator with index n ~ J1(A+). We may conclude similarly that (4.27)

8(A+)

= {SA

E F(Xj Y)

I A(+oo) = A+}

is a contractible space such that we are able to equip the determinant bundle Det on 8(A+) with a well-defined orientation. This concept of a one-sided operator in the trivia! framework can be transferred to the manifold M as follows. Let s E coo([O,oo),M) have the fixed end s(oo) Ax E E be fixed. Then we consider the pair (s, SA), where (4.28)

SA: H 1,2(S*TM1[0,00))

--+

L 2(s*TMI [0,00))

u

f-+

V'tU

x and let

+ A· u

is a Fredholm operator of type (4.26), endowed with the fixed end A(+(0) = Ax. This set {(s, SA)} of one-sided operators with fixed end Ax may be assembled in a uniformly orientable equivalence class [SAxl. The uniform orientability is due to the fact that, by analogy to Definition 3.10, the choice of orientation of an arbitrary representative (s, SA) together with a trivialization of s*TM induces an orientation of 8(A; triv)' Hence in a reverse and unique manner, this induces orientations for all'the other representatives, so that they do not depend on the actual choice of trivializations. This may be concluded by means of a lemma which is strictly analogous to Lemma 3.8. Moreover, it proves to be consistent with the fact that the space of all curves 8 with a fixed end s( (0) = x is contractible to the constant curve s == x. Additionally, we are able to analyse a gluing operation for such one­ sided operators together with the 'two-sided' operators (u, K) E ~u'TM of the former type: [SAxJ#[U, K] = [SK+J for Ax = K_ . Starting from a given coherent orientation with respect to the classes of op­ erators [u, KJ E A, we derive a unique orientation of [SK+] from a given orientation of [SAxl by means of this gluing operation. In order to obtain uniqueness guaranteed, we obviously need independence from the chosen curve u associated to the fixed endpoints. This is the very point where we remark the necessity of a simultaneous orientability of all operators (u, K) belonging to given ends K;, K:. Let us state the consequence of this uniqueness as

opemtors5

Proposition 4.10 A coherent orientation of the relative classes of [K;, K:J together with an orientation of a given fixed [SA xl , Ax E E induce 5with respect to the equivalence of orientations by means of trivializations with coincidence at the negative end

~'"

. . • _A _ _ • _ _

~~~~~~~~~;;~·\!~~~~iEi~~,'t~t!¥iiL~:~J.~~~...!

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

159

unique orientations of all positively one-sided classes of opemtors [S K:], K: E

E. On the other hand, however, let us regard what we have to change, if we wish to endow the classes [SK-], K; E E of negative half-sided operators with a uniform orientation, that is in principle an orientation of the unstable manifolds. In the case of a non-orientable manifold M we would also have to change the definition of equivalence of the relative operators (u, K) rv (v, L). Namely, we would have to choose the coincidence of the trivializations of u*TM and v*T M at the positive end u( +(0) = v( +(0) as far as 'n-dimensional Mobius bands' WI (( u· V-I )*T M) -=F are concerned. Actually, the question of whether we have to endow either the 'stable' or the 'unstable' manifolds with a uniform orientation corresponds to the decision between negative and positive gradient flow. It is exactly one of the two alternatives, namely the choice of the stable manifolds in the case of the negative gradient flow, which reproduces the necessary identity for the classical homology of connected manifolds (see below), Ho(M) 9:! Z, also in the situation of a non-orientable manifold.

°

Proof of proposition 4.10. Without loss of generality, let s E Coo ([0,00], M) and Ul, U2 E Coo(iR, M) be asymptotically constant and satisfy the identities

s(+oo) = Ul(-oo) = U2(~00) = x and Ul(+oo) = U2(+00) = y . Moreover, let us assume a positive half-sided operator Sx on s together with relative operators K 1, K 2 on Ul and U2, respectively, once again asymptotically constant with ~ K+ - -KSx I2 ' K+ 1- -K+ 2'

We consequently consider asymptotically constant trivializations

¢>x: s*TM ~ [O,ooJ x IR n , ¢>1 : urTM ~ iR x IR n , ¢>2: u;TM ~ iR x IR n satisfying ¢>x(+oo) = ¢>1(-00) = ¢>2(-00), such that the pair (¢>1,¢>2) is ad­ missible and the non-degenerate, conjugated self-adjoint endpoint operators6

Ax

=

¢>x(Sx)(+oo) = ¢>1,2(K1,2)(-00), A y = ¢>1,2(K1,2)(+00) E GL(n,lR)

are diagonal matrices. If the identity ¢>1 (±oo) = ¢>2 (±oo) holds at both ends, the proof is straightforward. Thus, let us assume the actually interesting case 6¢x(Sx) = ¢xo SxO ¢;l E .c(X; Y), see 4.26

CHAPTER 4. MORSE HOMOLOGY THEORY

160

¢l (+00)

=

4.1. THE MAIN THEOREMS OF MORSE HOMOLOGY

So ¢2( +00), where S denotes the reflection matrix

S~C'

+1.

161

o Before we are able to relieve the homology groups Hf(M) from their parameter a, we first have to introduce even an additional parameter. Choosing any class [SA], A E t:, of half-sided operators with fixed positive end and specifying any orientation O(SA) of this class, we expand the parameter dependence,

+1)

Ha ~ H(a,o(SA))

from Definition 3.7. Let us henceforth use the following notations:

= So ¢x, = So ¢l, = SO¢2 .

¢t = ¢x, ¢; ¢t+ = ¢l, ¢1 ¢!- = ¢2, ¢2+

Wecompute (¢;#o¢2+)(+00) = (¢t#o¢t+) (+00). Now let [SxJ be oriented firmly by o(Sx) as well as K 1 '" K2 by o(Kt} and o(K2). Then, the relation between the orientations (4.29)

o(Sx#O Kt} == o(Sx)#o(K1 )

~

o(Sx)#o(K2) == o(Sx#O K 2)

* * . 2 For instance, if we choose the Hessian A = H /(x), x E Crit I for a given Morse function I, specifying the orientation O(SA) amounts in principle to fixing an orientation of the stable manifold W 8 (x). Hence, the coherent orientation a gives rise particularly to an orientation of connecting orbits Ux,y E M!,y = WU(x) rh W 8 (y) thus leading to orientations of all the other stable manifolds. Now, let (a,o(SA)) and (a',o'(S~)) be pairs of parameters, such that there is exactly one transformation pair ±ip (see above). It is due to the coherence of the given orientations and to Proposition 4.10 that we may start without loss of generality from identical classes [SA] = [S~l. Let us choose the sign of ±ip according to the case distinction

amounts to relating (4.30)

(¢t#o¢t+)(o(Sx#O Kt}) ~ (¢; #o¢2+)(O(SX#O K 2))

on the contractible class 8(A y ) in trivialized form. Thus, we have to verify that this relation is equivalent to the compatibility of the orientations o(Kt} ~ o(K2 ) according to definition 3.10, that is (4.31)

¢t+(o(Kt}) ::: ¢!-(o(K2))

in

9(A x ,Ay) C ~triv

,

since (¢t+, ¢!-) is admissible. 7 We notice that we may state relation (4.30) in the trivialized case alternatively as (4.32)

¢t(o(Sx))#¢t+(o(K1 )) ~ ¢;(o(Sx))#¢2+(o(K2)) .

Now let L t be any homotopy between ¢t+(K1 ) and ¢2+(K2) in the con­ tractible space 9(A x , A y ). Given a large enough gluing parameter, it induces a homotopy ¢t(Sx) #L t between ¢t(Sx)#¢t+(Kt} und ¢t(Sx)#¢2+(K2). Then the identity

¢t (o(Sx)) = (S 0

¢;) (o(Sx)) = -¢; (o(Sx))

together with the equivalence of orientations (4.32) imply the relation (4.33)

¢t+(o(Kt}) ~ -¢2+(o(K2))

7See Definition 3.7.

=

¢!-(o(K2)) .

(4.34)

+1, if O(SA) = o'(SA) ip(A) = { -1, if O(SA) = -o'(SA)

To put it in other terms, we extend the action {±1 }£ x CA

----t

CA

to

ip. (a,o(SA)) = (ipa, ip(A) . O(SA)) , so that we deduce the existence of a now unique transformation map ip E {±1}£ between (a, O(SA)) and (a', o'(S~)) from the coherence condition for a and a'. At last, due to the construction, this mapping

((a,o(SA)), (a', o'(S~)))

1--4

ipa' a

satisfies the functorial property (4.35)

ip0'30'2 0

ip0'20'1

ipa30'1, ipO'O'

id,

which was demanded in (4.25). Thus, coherent orientations give rise to isomorphical homologies, u'u which may be identified with each other by means of the isomorphisms q)'P in a unique way due to (4.35). As a consequence, this identification process provides us with homology groups Hk(M) which are finally independent of any other parameters than the order k and the smooth manifold M.

,~::;ii;;;f}E#"~;;f;.~;;;"';;,i:;:;'~;;;~,ffi.7,;)t'&~;:;':"':::;,;i'&:, ..-;:.:tii';:.:'''',;;;~;;;;;;";;';;':';;::";'·"'o1t,~;:~;;;-;:;:j;;i~~~:~f,;,"';f.:f·~-~;;.;:;;;.,.",~;.~:t,':1~T;a;;;'~';;;:';'~,1.i>'j;."1~1t;,-~,;;;,:;;';;~"",.;,,,:,:~

162

CHAPTER 4. MORSE HOMOLOGY THEORY

Definition 4.11 By analogy with definition

4.8 we

define the Morse homology

groups as the inverse limit

Hl:°rse(M)

THE ElLENBERG-STEENROD AXIOMS

4.2.

gives rise to an admissible pair with respect to (ul,D u1 ) ,....., (u2,D u2 ). More­ over, we may construct this trivialization in accordance with the splitting-off of a one-dimensional subbundle rJ of ~ = ,*TM,

lim Hf(M)

=

~

+­

with respect to the isomorphisms 'P~' a .

Referring back to definition 3.7, our argument for the construction of the non-trivialized Fredholm-classes on the one hand was founded on the relation for the halfsided operators as described in proposition 4.10 and on the other hand on the algebro-topological relation Ho(M) ~ Z for connected man­ ifolds. The discussion of the latter, which is equivalent to the dimension axiom within the axiomatic framework is now anticipated to the next section. We verify the consistence of our orientation construction with this crucial relation as follows. Proposition 4.12 Let M be a connected, smooth manifold equipped with a Morse junction f, which possesses exactly one local miminum yo. Then, the Morse homology group of order 0 with respect to f satisfies the relation

The proof is based on the following fact:

(Ul'

any coherent orientation

(J

Ul

and U2 from Mty

DuJ ,. . ., (U2' D U2 ) ,

leads to the identity Ta(Ul)

=

= rJ EB C- l

.

Here, rJ denotes the one-dimensional bundle on [-1,1] induced by TWU(x) with the fibre rJo = TxWU(x) which is identical with the eigenspace of H 2 f(x) associated to the unique negative eigenvalue. Up to an appropriate parame­ trization, the generating sections Ul, U2 for the kernels of D U1 and D U2 , respectively, lie exactly in this subbundle rJ. Thus, starting from a fixed orien­ tation on (n-l, we can derive orientations of the bundle ~ = (u 11 . U2) * T M from the canonical orientations [Ul] and [U2], which must be opposite each other. This proves the lemma. 0 Proof of the proposition. The unique local minimum Yo of the Morse function f provides us with a generator of the cycle group Zo(f) for the Morse complex of f. This generator is unique up to sign. We deduce from the lemma that, given any coherent orientation (J, the boundary 8 a Xl of each generator Xl E C l (f) vanishes. Thus, the assertion follows from Bo(f) = {O}. 0

It seems natural to expect that the proposition remains true if we do not impose the special condition on f to possess only one local minimum. The proof in this general setting requires some more steps and will be given in the next section in relation to the dimension axiom.

Ho(f) ~ Z

Lemma 4.13 Given two different, isolated trajectories with (x,y) E CritIf x Critof, that is

163

-Ta (U2)

Proof We consider the connection of the curves at the point x referring to a

suitable reparametrization which yields , = u 1l . U2 E Coo ([-1,1], M)

This is nothing else than a suitable parametrization of the I-dimensional unsta­ ble manifold WU(x) as a I-cell. We now choose a trivialization of ,*TM, such that on the one hand the Hessians H 2f(x) = D;;l = D;;2 and H 2 f(y) = D;;l = D;;2 become diagonal matrices with respect to this trivialization, and such that on the other hand the induced isomorphisms TyM = T,(±l)M ~ lR n differ by the reflection S defined above if they differ at all. Hence, this trivialization

4.2

The Eilenberg-Steenrod Axioms

In this section, we shall carry out the final step of the development of Morse

homology theory. We shall verify the functorial properties and the accordance with the axioms of a homology theory as they were set up by Eilenberg and Steenrod in [E-S], namely, • the existence of a long exact homology sequence, • the homotopy axiom, • the excision axiom and • the dimension axiom. For the sake of simplicity, we shall first derive the functorial construction and homotopy invariance for the concept of the absolute homology groups H~orse(M), before we treat the somewhat more involved relative groups H~orse (M, A) associated to admissible pairs of manifolds (M, A). In this framework

..",;:.:;;;;::,,~

CHAPTER 4. MORSE HOMOLOGY THEORY

164

a natural induction of chain maps for Morse complexes by closed embeddings of manifolds proves essential for functorial construction. In order to discuss this construction, however, we first need some technical preparation, namely suitable extension lemmata for Morse functions. 8

4.2.1 Extension of Morse Functions and Induced Morse Functions on Vector Bundles

4.2. THE ElLENBERG-STEENROD AXIOMS

we define (4.36)

fn = f/AnB n

165

.

Consequently, there is an increasing sequence of positive numbers (rn)nEN C JR+, such that the condition

fn(A n B n ) C [0, rn] is true for all n EN. This enables us to apply Tietze's theorem inductively for n: For each n E N there is a

The problem of the extension of Morse functions from submanifolds of M to the whole manifold M may be separated into a technical one on the one hand, the extension of the coercivity property and of the regularity of Morse functions on open submanifolds, and on the other hand, a problem related to dimension with respect to closed submanifolds. The first lemma yields the extension of the coercivity property. We notice that the coercivity of a continuous function f : M ----7 JR, Le. M

a

= {x

E M I f (x) ~ a }

compact for all a E JR ,

°

Lemma 4.14 Let M be a smooth manifold, A c M be a closed subset and f : A ----7 [0,00) be continuous and proper. Then there is a continuous and proper extension 9: M ----7 [0,00) 91 A = f . Proof. The proof is basically founded on an application of Tietze's extension theorem for continuous functions:

Let X be a normal topological space, A c X be a closed subset and f : A ----7 [0, 1] be a continuous function. Then there is a continuous extension

1* : X

----7

[0,1]' 1*1 A = f .

Since we are able to embed M as a closed submanifold in an JRN with N chosen suitably large, we may start from the assumption that M = JRN without loss of generality. Denoting the closed ball with radius r by Br

=

8for coercive Morse functions!

{x E JRN

I Ilxll

~ r}, r >

°

an d

o 9nl B n- 1

_

-

90

n-l

Defining

h n = arctan9~: B n

----7

[0, ~), n EN,

we obtain a sequence of continuous functions once again according to Tietze's theorem,

k n : JRN

There is a constant c > such that the continuous function fe = ----7 [0,00) is non-negative and proper, Le. f;;1 (K) is compact in M for all compact K c [0,00).

: B n ----7 [0, r n] satisfying

9~IAnBn = fn

is equivalent to the following property:

f +c : M

9~

(4.37)

.

----7

[o,~] with knl B 2 n

= h n and

lim k n = ~2 . Ilxll-->cx>

It is due to the construction in (4.36) and (4.37) that we may state the uniform. convergence on compact sets,

(4.38)

kn

Sh k E CO (JR N , [0, ~2))

with

lim k(x) IIxll-->cx>

= ~2

This implies a continuous and proper extension of f by 9 = tank E CO (JR N , [0,00)),91 A =

f o

We now have to transfer the extension result to the dense subset

CCX>(M, JR) c CO(M, JR)

of the smooth functions. Furthermore, we have to add the regularity condition

df rh McT*M.

We refer essentially to the results of differential topology as they can be found

in [Hi]. Hence, we take over the notations C~(M, JR), NU, f), etc., for the strong or, equivalently, the Whitney topology and the basis sets on Cg(M, JR)

NU, f) = {9 I 19(x) - f(x)1 < f(X) for all x EM}, f E CO(M, JR+) . An analogous definition holds for C~(M, JR). Then the extension result for smooth Morse functions can be stated as

CHAPTER 4. MORSE HOMOLOGY THEORY

166

Lemma 4.15 Let M be a smooth manifold, let A be a closed and W an open subset and let f E COO(W, R) be a Morse function on the submanifold W, such that Crit f cAe A eWe M .

Then there is a smooth Morse function 9 on M which extends f, that is,

4.2. THE ElLENBERG-STEENROD AXIOMS We consequently define

go: M -t R, gOI U

=

f U . l

The set of continuous and proper functions represents an open subset with respect to the Whitney topology (see [Hi]), Prop~(M, R) C C~(M, JR.), open subset .

.AI,(go)

C

Prop~(M, JR.) C C~(M, JR.) ,

N.(go)

c

C~(M,JR.)

with

Crit h n U\A = 0 for all

Thus, the intersection .AI = JR.), too.

Nt n.Al,

hEN.

forms an open and dense subset of the Baire space C~(M, R). Since COO(M, JR.) lies dense in C~ (M, JR.), too, there is a

Now let

Coo (M, JR.) n.AI n X

0: :

arbitrarily close to go in C~ (U\A, IR)

M -t [0,1] be a smooth function satisfying

0:, A =

° and

0:1

and

gl A

=

fl A ' 0

As pointed out above, the next step is to develop an extension result for closed submanifolds. Here, we require an additional construction method for Morse functions, which appears very naturally. A feature which proves crucial for the whole Morse homology is the canonical tensorial behaviour of Morse complexes with respect to product operations: Given two manifolds equipped with Morse functions (M, J) and (N, g), the operation

(J EB g) (m, n)

(4.43)

= f(m)

+ g(n)

naturally provides a Morse function E COO(M x

N,IR)

together with the canonical identification (4.44)

CritkU EB g) =

U Critd x Critjg EM x N

,

i+j=k

At this stage, regarding our current concern, we need a generalization toward smooth vector bundles

is an open and convex subset of C~(M,

X = {f E C 2 (M, JR.) I df rI1 Me T* M }

E

.

because the Morse index behaves additively with respect to the operation EB.

It is clear from the well-known transversality theorems in differential topology (see [Hi]), that

g'

+ 0:' g'

provided that giU\A has been chosen close enough to gOIU\A'

fEBg

which contains merely coercive functions. Furthermore, referring to the C2_ Whitney topology, we can find a convex, open neighbourhood (4.41 )

go

Due to the concrete definition of the convex neighbourhoods N.. and the identity of go and f when restricted to U, 9 satisfies the properties

Therefore, we find a convex neighbourhood (4.40)

0:) .

9 E COO(M, R) n.AI n X

Due to Lemma 4.14 there is a continuous coercive extension (4.39)

9 = (1-

(4.42)

glA = flA . Proof. Since M is a normal topological space, we find an open subset U C M satisfying AcUcUcW.

167

M\U = 1 .


Let (".) E be a Riemannian metric on E with the associated quadratic form

q(Vm ) = (v m , vm)m

and let f be an arbitrary Morse function on M. Then we obtain the Morse

function IE:E-t1R (4.45) fdv m ) = f(m) + q(v m ) on E with the property (4.46)

lEI M

=

f and Crit*IE

~ Crit*f .

~~:~~~~~~~-~~~=-;-::~;:r.~:..~:~------"

~;:."~'" -'~-':.,-,:~~~::~o;:.·~·_,,,,~c~~:~;~~.=~:§~=~;;:~~~~~':g:s:}:ff;:::===~= -..--==-~~~~~=~=:~~;~~~~:;:;:~~~~~~:=~~~~~~~-;:-;':':~;;~~~-E:==:=~~~~~:~:~~~r~-::~"t'"~;':S~:.::;~¥:~F,~;~§=~~~~~~=!.:~_~A5:¥~~~~·· __-';' .r"-.,'.

168

CHAPTER 4.. MORSE HOMOLOGY THEORY

On the one hand, the latter is due to the local product structure of E, so that the results (4.43) and (4.44) may be transferred to this special bundle situation. On the other hand, we observe that q restricted to the fibers is a trivial Morse function, that is Crit.qm = Critoqm = {Om} .

4.2. THE ElLENBERG-STEENROD AXIOMS

169

Proof. It is due to construction by means of quadratic extension on the nor­ mal bundle 1/ that the negative gradient field -\7 Iv represents an inner nor­ mal vector field on the boundary q; 1([0, EJ) of the manifold with boundary q;l ([0, ED for any E > O. This means that -\7 Iw is an inner normal vector field for A< in A<. From this we obtain the corollary. 0

a

a

This feature enables us to derive the following extension result:

Proposition 4.16 Given a closed submanilold A C M and a Morse junction IA E COO(A, IR) on A, there is a Morse junction I E COO(M, IR) extending

lA,

IIA = IA . Proof. Having in mind the above preparations for naturally induced Morse functions on smooth vector bundles, we consider a tubular neighbourhood W of A, which always exists for any closed submanifold (without boundary): 'P:

\}WCM.

4.2.2

The Homology Functor and Homotopy Invariance

The Diffeomorphical Functor First, we wish to discuss the homomorphisms between the respective homology groups induced by diffeomorphisms. We assume 'P: M --2..., N to be a smooth diffeomorphism between two manifolds and I to be a Morse function on M. Obviously, 'P gives rise to the Morse function

~

1/_

'P.I =

(4.48)

1 0 'P- 1

on N, so that we obtain the one-to-one correspondence

A Here, 1/ is the normal bundle of A and W is an open submanifold. Thus, the diffeomorphism 'P induces a Morse funct~on on W,

Iw = Iv 0 'P-

(4.47)

1

,

which extends I A : Iw I A

= I A· The preconditions of Lemma 4.15 are fulfilled if we choose a suitable restriction: Let R > 0 and AR = 'P (q;l ([0, RJ)), that is,

AR

C

II AR+< = IWI A R +< that is, in particular, IIA

= IA.

I

on M satisfying

' o

Actually, we may gain a further important result from this special extension construction on closed submanifolds:

Corollary 4.17 The extension I 01 I A in the above proposition can be chosen in such a way that there are no trajectories lor the negative gradient flow 01 I leaving A.

'P : Crit. I ~ Crit. ('P. f) .

However, before we are led to a chain map between the associated Morse com­ plexes 'P. : C.(I) ----> C.('P.f) we have to verify the condition (4.50)

A R +< eWe M, A R closed .

Then, Lemma 4.15 gives rise to a Morse function

(4.49)

a 'P.

= 'P.a .

Provided that the generic Riemannian metrics on M and N have already been fixed, it is generally merely an isometry 'P which maps isolated trajectories for I onto isolated trajectories for 'P.f. But we are able to choose the generic metric g on M in a way that also provides a generic metric on N via in­ duction by 'P. This means that we may choose the metrics generically such that 'P becomes isometrical. Remember that the transformations of the Morse complexes associated to fixed Morse functions by means of suitable Morse ho­ motopies. Analogously, we may regard the homology groups as independent of the Riemannian metric and we therefore assume without loss of generality that the given 'P is isometrical. The only item left to verify with respect to (4.50) is the consistency of the coherent orientations which have been chosen for M and N independent of 'P. This can be achieved by means of an appropriate

170

CHAPTER 4. MORSE HOMOLOGY THEORY

transformation by an a E {±1}Crit.! as we described in the last section, that is (4.51)

'P. : C.(J) ~ C.('P.J) 'P.Xk = a(Xk) . 'P(Xk), Xk E Critk!

To sum up, up to a respectively appropriate homotopy of the Riemannian metric each diffeomorphism 'P: M ~ N induces a homology isomorphism

'P~ : H~M(M, J) ~ H~N(N, 'P.J)

!:.4 ff3

'P. f

'P.

ff3

Vh t .-----,====

Jl + Iktl21Vhtl2

(4.52)

ifJ.('P.f ,'P.r)

0

'P. = 'P.

0

'P. : C.(M, J)

0

according to the definition of H.(M)

~

'P. : H.(M, J)

'Y

ifJ.(Jf3, r)

H.(N)

= limH.(M, r). Obviously, the con­ +-­

('P 0 'l/J). = 'P.

0

C.(N, 'P.J)

~

H.(N, 'P.J)

and

'P. : H.(M)

=

struction guarantees consistency with the compositions of the morphisms (4.53)

~

and consequently to the homomorphisms

implies that 'P in fact induces a homomorphism

'P. : H.(M)

--=:.. 'P(M)

gives rise to the chain map

,

are mapped by 'P onto the corresponding h-trajectories belonging to 'P.ht h t 0 'P- 1. In a nutshell, the equality

f3

arp'/I C.('P~J) = a'P~!

(4.54)

'P : M

such that in the case of an isometry the h-trajectories associated to

1'=

N. We henceforth denote by 'P~f the Morse function on 'P(M) which is induced by f and 'P and which can be extended to a Morse function 'P.f on the whole manifold N without any negative gradient flow trajectory leaving the submanifold 'P( M) according to the corollary of Proposition 4.16. Actually, this condition allows us to identify the Morse complex C. ('P~J) on 'P(M) with a subcomplex of C. ('P.J) on N, that is

,

we can confirm functorial consistency. 'P induces a homotopy a rp.ht -----+

171

Here we assume a suitable extension of the coherent orientation of 'P(M) to the whole N. Thus, the identification with the subcomplex together with the diffeomorphical functor with respect to the diffeomorphism

Now, regarding a homotopy of Morse functions

r

4.2. THE ElLENBERG-STEENROD AXIOMS

'l/J. ,

so that we obtain a functor as far as smooth manifolds together with diffeo­ morphisms are concerned.

The Embedding Functor and the Homotopy Lemma By means of the extension results and this diffeomorphism functor we are now able to accomplish the extension to closed embeddings. Let 'P : M '----> N be an embedding such that 'P( M) represents a closed submanifold within

~

H.(N)

with respect to the identification process. Once again, we verify the composition rule, so that we obtain the sequence of functors

(4.55)

Hk

:

(manifolds, closed embeddeddings) ~ AB,9 kENo.

Before the next step of the generalization aiming at arbitrary smooth mappings, we first prove the crucial

Lemma 4.18 (homotopy lemma) A smooth i-parameter family of closed embeddings ('Pt : M '----> N)tE[O,lj implies the identity

'Po. = 'Pl. : H.(M) ~ H.(N) The proof of this lemma can be reduced to the following special case:

Auxiliary Proposition 4.19 Let f be a Morse junction on M and i O, i l M '----> lR x M be the embeddings given by

i"(m) = (v, m) for v = 0,1 9category of Abelian groups

~~TI~~:~~L~~{--------=-:~~~~~i..~~~~~~~~~1

172

CHAPTER 4. MORSE HOMOLOGY THEORY

Additionally, we consider the quadratic functions qv : JR

JR qV(X) = (x --+

V)2, V

= 0, 1

Then the isomorphisms

.

i~ : H.(J) ~ H.(qv EB I)

induced by the canonical identification Crit.f ~ Crit.(qvEBI) and the canonical homology isomorphism

ep~o : H.(qo EB f) ~ H.(ql EB I) fit in with the commutative diagram

z.o

H.(J)

Y

H.(qo

+ I)

EB

eplO

H.(ql

f

Conclusion: If we carry out the identification of the homology groups H.(J) for Morse functions j on JR x M, we obtain the identity

173

and we define the homotopy

ht : JR x M

(4.58)

--+

JR, t E [0,1]

ht(x, y) = (x - a(t))2

+ f(y)

It is clear that this 'translation' represents a Morse homotopy between qo EB f and ql EB f. For the sake of simplicity we content ourselves with a consideration of the homomorphism epZt with respect to the isolated h-trajectories for the time-dependent negative gradient flow

(t3(t), i'(t)) = -Vh t

(4.59)

0

(/3(t), ')'(t)) . 1

We leave out the norming by the scaling factor (1

+ Ihtl2lVhtl2) -2.

This is

admissible for the following reasons: First, we will derive explicitly the necessary compactness results for this special Morse homotopy. This feature was the only reason in general for the scaling. Second, by means of a homotopy using an additional parameter A, we may verify that the homomorphism epk on the level of homology is independent of the special form of this norming. The isolated h-trajectories

•

~) + z.

4.2. THE ElLENBERG-STEENROD AXIOMS

(/3, ')')

E

MZ~,Zl' zv

= (v, y~)

E

Critk(qv EB I), v

= 0, 1

are determined uniquely by (4.59) as solutions of the differential equation

(4.60)

(t3(t),i'(t)) = (2(a(t)-/3(t)),-Vf o ')'(t))

Now let \11 be the isomorphism

(4.56)

i~

=

i~ : H.(M)

--+

H.(JR x M) \11 =

from this commutative diagram.

Proof of the auxiliary proposition 4.19: We have to verify the existence of a Morse homotopy

~ L 2(JR, JR,) ata + 2 : H'1 2( JR, JR ) ~

\11 E E triv

Since it is a smooth function with compact support according to the construc­ tion, we observe that

ht

qo EB f ~ ql EB f ,

it E CO'(JR, JR)

such that the associated homomorphisms epZt : Ck(qo EB f) --+ Ck(ql EB J) map the generating critical points to each other according to the formula

As a consequence, \11-1 (it) is determined uniquely. Thus, /30 = C<Xl(JR, JR) solves the equation

(4.57)

epZt (0, Yk) = (1, Yk)

for all Yk E Critkf, kEN .

(4.61)

(:t + 2) .

c L 2 (JR, JR) .

/30 = \I1(a) - it =

20:

We consider a smooth, monotone function a E C<Xl (JR, [0,1]) satisfying To put it in simpler terms,

a(t) = { 0, 1,

t~O

t~O

.

f30

is the unique solution of

t3o(t) = 2 (a(t) - /3o(t)) , t E JR j /30 E 'Pt:;(JR, JR)

0: - \11-1 (it)

E

""""""=========~~~~~~~~~~~~~~~~""""",~-"",~"""~~"""-~-~,,,,,,~~,,,,,~~,,=~_.~=~-~--~:"""""""""""",,,,,,,,,,,,~~-,,=,~ ~ """'~~

174

CHAPTER 4. MORSE HOMOLOGY THEORY

This proves the statement that each isolated trajectory ((30, /,) E M~~,Zl cor­ responds to exactly one trajectory of the time-independent negative gradient flow for f. This amounts to the one-to-one correspondence f M x,y /'

s.<

~ f->

({30, /,)

I yk E Critk/ }

N(

IN

X

Critj(qk EB I), j = 0, ... , n,

qk(X) = (x, x), x E IRk First, we again obtain the isomorphisms

iZ : H*(J) ~ H*(qk EB I)

i: :H*(M) ~ H*(IR k

P: =

(4.63)

• IR

~

X

yielding

M)

independent of the Morse function f. This is due to compatibility with Morse homotopies. Thus, the inverse mapping by means of the identification of critical points represents the homomorphism pZ induced by the projection map pk,

[~

·t

Critj(J)

0

• IR x M

EB

which play an essential role second to that of closed embeddings. The idea is to extract the functorial behaviour of the homology morphisms step by step from the projections pk : IRk X M ---> M. By analogy with the I-dimensional case in the auxiliary Proposition 4.19 we consider the canonical identification

with respect to the canonical quadratic form

Proof of the homotopy lemma. Let us consider the commutative diagram of closed embeddings for any t E [O,lJ:

[

In the next passages we concentrate on projection mappings of the type p:MxN--->N,

From this we easily conclude the assertion.

~,

The Homology Functor for Arbitrary Smooth Mappings

ZO,Zl'

M~~,Zl = { ({30, yk) E P~~~Zl (IR x M)

Me

175

4.2. THE ElLENBERG-STEENROD AXIOMS

Mh,

with Zo = (0, x) und Zl = (1, V). Due to the assumption we are considering the relative Morse index /l(x) = /l(zo) = /l(Zl) = /l(Y). Thus, the time­ independent f-trajectories in question have to be constant, such that M~ot , z 1 consists exactly of the trajectories

·t 1M

--~-----~,--~ '---.---~~=-~-~-~~~--~' -~-~~=---~~~~"-

N

(i:r

1 :

H*(IR k

X

M) ~ H*(M)

Using the notation X

M

'--+

IR k+1 X M

((Xl, ... ,Xk),

m)

f->

((Xl,,,,,Xk,O, ... ,O),

ik+l,k : IRk

where

¢>(t)

(t, 'Pt(X)) ,

=

we may immediately set up the commutative diagram

i~(x) = (t, x), i~(y) = (t, y)

H*(M)

Y

Since we are already allowed to deal with the covariant homological functor for closed smooth embeddings, we deduce the identity (4.62)

.-/,.·t·t '+'* 01M* = IN*

0 'Ph

( ) : H* M ---> H*(IR

X

N)

independent of t E [0,1], also.

~k+l 1*

1*k

(4.64)

The auxiliary proposition states that isomorphisms i~* and i~* do not de­ pend on the parameter t. Hence we obtain the identity 'Ph -

m)

H*(IR k

X

M)

EB .k+l,k 1*

•

H*(IR k+1 X M)

~ffi~ H*(M)

iN:;-lo¢>*oiM*

o

which describes the relations between the inclusions and the projections as far as the trivial extensions by IRv are concerned.

176

CHAPTER 4. MORSE HOMOLOGY THEORY

Definition 4.20 Resorting to a suitable closed embedding cpk : M ~ IRk we are able to lactor the projection map p : M x N -+ N into

p = pk

A-. _ 'l't -

=

p~

0

(cpk

X

idN

L '.

Let 'l/Jl : M ~ IRI be another closed embedding. Extending both given embeddings by means of the inclusions ik+l,k and ik+l,l, i.e. 0

cpk and ik+l,l 0 'l/Jl : M

~

IR k +1

.

Summing up, it is sufficient to find a homotopy

(4.65)

cp

~ 'I/J, ¢t: M ~

IR

2n

¢I : M x [0, 1J

-+

((1 -

T(t)) . cp, T(t) ,cp)

¢2 : M x [0, 1J -+ IRn EB IR n , ¢2(., t) = (T(t). 'I/J,

(1- T(t)) ,cp)

ic

(cp, 0)

(0, cp)

('I/J,O) .

:

N

y

Obviously, ¢t fulfills the

0

i c•

= idH.(N)

~

M x N,

f--t

(c, y), c E M fixed,

from the homotopy lemma. Now we are able to associate to any arbitrary smooth mapping be­ tween manifolds a homomorphism between the respective Morse homology groups.

Definition 4.21 Let IE COO(M, N) be a smooth mapping. Then the graph­ embedding (id, f) : M ~ M x N

represents a closed embedding, so that

(id,f).: H.(M)

-+

H.(M x N)

is well-defined, Resorting to the projection homomorphism constructed above, p. : H.(M x N) -+ H.(N), we now define

(4.67)

I.

= p.

0

(id, f). : H.(M)

-+

H.(N)

To sum up the functorial construction, we choose indirect way toward the gen­ eral homological functor via the factorization I = p 0 (id, f) into mappings which can be treated more easily. We now have to verify that this defini­ tion is compatible with the already existing functor for closed embeddings and projection maps and that the cofunctor property of a covariant functor

Thus we obtain smooth homotopies of closed embeddings for each fixed param­ eter t satisfying 4>~ ~

T.

for arbitrary inclusions

IRn EBIRn ,

and

4>~ ~

p.

closed embedding for all t E [0,1]

Let T : [0,1] -+ [0,1J be a smooth monotone mapping, locally constant at 0 and 1 satisfying T(O) = 0 und T(l) = 1. Then we define the homotopies

¢I(., t) =

(4.66)

,

and regarding the functorial behaviour of the homological morphisms with respect to closed embeddings together with relation (4.64), we may start from two closed embeddings cp, 'I/J : M ~ IRn without loss of generality. Due to the homotopy lemma and the uniform product structure it is sufficient to construct a homotopy cp ~ 'I/J. Since we are already allowed to exploit the functorial behaviour with respect to the composition pn,2n 0 i 2n ,n = idIRn, we are free to compose embeddings cp and 'I/J with the inclusion i 2n ,n : IR n ~ IR 2n

¢~t, 0 ~ t ~ 4 2 1 ¢2t-I' 2~t~1

By means of this method of factoring through a closed embedding cp : M ~ IRk we may derive the identity

Obviously, we have to verify that this homomorphism p. does not depend on the noncanonical choice of the embedding cpk, For this item we rely on the homotopy lemma.

ik+l,k

{

as a smooth homotopy due to the definition of required properties.

p. : H.(M x N) -+ H.(N)

177

We finally reparametrize this composition of homotopies obtaining

(cp'" x id N )

0

According to this lactoring we define

p.

4.2. THE ElLENBERG-STEENROD AXIOMS

(g

is satisfied.

0

f). = g.o I.

~

178

~:;._

CHAPTER 4. MORSE HOMOLOGY THEORY

Auxiliary Proposition 4.22 Given a closed embedding f : M '--+ N, the directly induced homomorphism (CPI). from (4·55) is identical to f.:

f.

= (cp I).

: H.(M)

---->

(4.68)

(id, fl

•

_~ __ ,;~_=

.'"""ijt---_::~>n

_

n:::::.. . ,-'""iI!'1'!?""

179

4.2. THE ElLENBERG-STEENROD AXIOMS

Proof. At first, let us consider the commutative diagram N ((O,i~~ 1R7n x N ld

(id,g) 1R7n

X

N x P

X

=Z

[(id,91

(4.69)

l~

EEl

NxP

N

r (~i1) r~

M xN

.. o._~

2;; _

P2

f

@

rpM

_

H.(N) .

Proof. Let cpM : M '--+ 1R 7n be a closed embedding. We consequently obtain the commutative diagram M (

_=~"~2-

X

~d 7n 1R

X

p

N

• N .

The two closed embeddings

Due to the definition of i*, p*, etc., the following identity is true:

cPl=(cpxid)o(id,f)

can be related by means of a homotopy through closed embeddings

cPt = (t . cp, I) : M

---->

1R 7n

X

P2.

(4.70)

0

(id, g)* = P3.

0

(id

As to the projection map P~ : 1R7n x N

(id, g)).

X

0

i* .

N, we obtain the isomorphism

---->

i* = (p~*rl : H*(N) ~ H*(lR7n x N) and hence the identity

(4.71)

cPo=iof and

• P.

g* 0 P~* = P3*

(id x (id, g)). : H*(lR

0

m

x N)

H*(P)

---->

from (4.70) according to the definition g* = P2* 0 (id, g).. Setting P3 = m po (id X cpN x id) and choosing suitable closed embeddings cpM : M '--+ IR and cpN : N '--+ IR n we are led to the commutative diagram

M xN

N, t E [0,1] .

Then the homotopy lemma implies the identity cPo. = cPh from which the o proof follows immediately due to the equality p* 0 i* = id*.

(id,flJ

We are able to prove analogously

M

b?M x id

id x (id g)

'. 1R7n

• IRm x N (

EEl

p\

f

It;

N x P

lp3

EEl 9

• N

X

~

P

Thus, we deduce the identity

Auxiliary Proposition 4.23 The projection map p : M x N ----> N directly gives rise to the homomorphism (cpp)* from Definition 4.20, and this homo­ morphism is identical to p*:

p* = (cpp). : H*(M x N)

---->

H*(N)

The last step toward the covariant homological functor follows from

Auxiliary Proposition 4.24 Any two smooth mappings f E COO(M, N) and 9 E COO(N, P) satisfy the composition rule

(g 0 I). = g*

0

f* .

(4.72)

g.

0

f*

= P3* 0

(id x (id, g)).

(cpM x id).

0

0

(id, f).

from (4.71) together with this diagram. Altogether, we come to the factoriza­ tion

(4.73)

g.

0

f* = p*

0

cPo. ,

where cPo = (cpM, cpN 0 f, go f) describes a closed embedding. From the other direction, we consider the expression

(4.74)

(g

0

1)* = P*

0

cPt.

using the closed embedding

cPl = ('PM x id)

0

(id, go I) : M

'--+

1R7n

X

n

IR x P

~i!l~~~#?:i:~?fu:t~*~±i-efL~~l)! . ~if~~~.;::!ifbtl:l:};~sc~1i~'~~.i~~;,k~it{;~~If;,;~~:4tz~~~~%J.~4-'i%~~l$,~i:4f;@ii;§!,i14K4~et:'~~~=~~!t}~~L*G;~tJf;S;it4tij~!t~E;t!

180

CHAPTER 4. MORSE HOMOLOGY THEORY

Relying on (4.65) we may once again set up a homotopy ¢o ~ ¢l through closed embeddings ¢t : M '----+ (lRm x lR n ) 2, so that the assertion follows from the homotopy lemma. 0 Proposition 4.25 The Morse homology groups

Hrorse(M) = ~Hk(M, f), kENo can be organized as a family of covariant functors from the category of smooth manifolds without boundary together with smooth maps as morphisms into the category of Abelian groups. Moreover, homotopical maps induce identical group homomorphisms, i.e. H k is a covariant functor for each kENo H k :

(

COO -m'ds. homotopy classes) ~" of Coo-maps

(M J£L N)

--+

AB

I->

(I.: Hk(M)

--+

Hk(N))

Proof. We have developed all the details except the explicit homotopy invari­ ance for arbitrary smooth mappings. But it is due to the definition of f. by means of the factorization that this follows immediately from the homotopy between the closed embeddings (id, f) ~ (id, g) induced by f ~ 9 and from the homotopy lemma. 0

4.2.3

Relative Morse Homology

In this section we finally analyse the feature which is still lacking with respect to an axiomatic homology theory. That is the concept of relative homology groups associated to admissible pairs (M, A) of manifolds. The underlying idea is to deduce such homology groups from a chain complex, which is now generated by critical points of a Morse function f on M outside of A. The appropriate algebraic object is a quotient complex analogous to the singular theory, C*(J, fA) = C*(J)/C.(JA) .

In this situation, the Morse complex C. (J A) of a Morse function on A has to admit a canonical identification with a subcomplex of C. (I). Altogether, we once again require a functorial concept which guarantees at each stage compatibility with the axiomatically required long exact homology sequence. As to this concept of a relative homology theory, we first have to compile some technical preparations.

4.2. THE ElLENBERG-STEENROD AXIOMS

181

Relative Morse Functions and the Relative Morse Complex Definition 4.26 We call a pair of smooth manifolds without boundary (M, A), where A is a submanifold of M, admissible, if either A is already a closed submanifold or the topological boundary 8A is a l-codimensional, orientable, closed submanifold of M. If (M,A) is admissible and of the latter type, we call a smooth function f E COO(M,lR) steep w.r.t. 8A, if -Vf represents an inner normal field with respect to 8A, that is: • Vf rh 8A

c

M and

• the negative gradient flow of f on 8A flows into A.

Since we have already treated sufficiently the aspect of suitable Morse functions for closed submanifolds, we shall henceforth consider the situation of admissible pairs of manifolds where A is an open submanifold whenever we do not give an explicit specification. Remarks (a) The just defined steepness of a function with respect to 8A is obviously a convex property, that is: If f,g E COO(M,lR) are steep w.r.t. 8A, the same is true for all (1- t)f + tg, t E [0,1]. (b) If f is steep w.r.t. 8A, it holds that Crit f n 8A = 0. Moreover, since M is a normal topological space, there is a neighbourhood U(A), which separates A from Crit f, U(A) n Crit f = 0. Definition 4.27 We now call f E COO(M,lR) a Morse function on (M,A), if f is a Morse function on M and additionally steep w.r.t. 8A. In the situation of a closed submanifold A C M, instead of steepness, f has to yield a subcomplex C.(JIA) of C.(I) in the sense of Corollary 4.17, i.e. there are no f-trajectories leaving A. In principle this definition establishes the coercivity property for the Morse function f on M relative to A. Since due to the above assumption A is an open submanifold of M, fl A cannot be a Morse function on A if 8A i= 0 holds. In this case the coercivity property is no longer satisfied, as not all sublevel-sets are compact. lO However, since we are able to isolate the critical lOReferring to Definition 2.40 concerning the compactness analysis we restate that we have to consider manifolds with complete Riemannian metrics. Since the submanifold A with non-void boundary cannot be complete with respect to the induced Riemannian metric, we have to alter this metric near the boundary. Similar to the following argument, this can be accomplished without changing the Morse-homological information, i.e. Crit. (J A) and the connecting isolated trajectories.

182

CHAPTER 4.. MORSE HOMOLOGY THEORY

points of f from the boundary 8A, it is possible to transform fl A on a suitable neighbourhood of 8A to a Morse function fA such that the respective Morse complex, that is, the critical points and the associated trajectory spaces within A, remains unchanged. Given the closed submanifold 8A bourhood

~,

c

M we choose a tubular neigh­

"\/WCM

N~ = {v x E 1/ Illvxll < E(X) for all x E 8A},