1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

182 D. SALAMON, FLOER HOMOLOGY

which, for a generic homotopy H°'f3, is a smooth manifold of dimension

dim M(x°',xf3;H°'f3) = μHa(x°') - μmi (xf3)(mod 2N).

As before, the local dimension near u will be denoted by μ( u, H°'f3) and we write

M^0 (x°',xf3;H°'f3) = {u E M(x°',xf3;H°'^13 ) : μ(u,H°'^13 ) = O}.

This is the zero dimensional part of the moduli space. As in Proposition 3.1 one

proves that this is a finite set for a generic homotopy H°'f3. Counting the elements

of these sets with appropriate signs gives rise to a homomorphism

<J>f3°': CF*(H°')---> CF*(Hf3)

defined by

xf3EP(Hf3) uEMD(x"' ,xf3)

The next lemma shows that this is a chain homomorphism and hence descends to a

homomorphism of Floer homologies. The subsequent lemma shows that the chain

maps <J>f3°' satisfy the obvious composition rule (for catenation of homotopies) and

the third lemma shows that the induced map on Floer homology is independent

of the choice of the homotopy. The main technical ingredients in the proofs of all

three lemmata are Floer's gluing construction and Gromov compactness.

Lemma 3.10. The above homomorphisms <J>f3°': CF*(H°')---> CF*(Hf3) satisfies

[Jf3 0 <J>f3°' = <J>f3°' 0 8°'.

Proof. Examining the 1-dimensional moduli space M^1 (y°', xf3) one finds that

8M^1 (y°',xf3;H°'f3) LJM^1 (y°',x°';H°') x M^0 (x°',xf3;H°'f3)

x"'

U LJM^0 (y°', yf3; H°'f3) x M^1 (yf3, xf3; Hf3).

yf3

This equality is to be understood with appropriate orientations and hence is equiv-

alent to the assertion of the lemma. (See Figure 9.) D

Figure 9. Floer's chain homomorphism

Lemma 3.11. Let H°'f3 be a regular homotopy from H°' to Hf3 and Hf31 a regular

homotopy from Hf3 to H^1. Define

H°'' R ,s,t - { -

for s::::; 0,

for s 2: 0,