LECTURE 3. FLOER HOMOLOGY 183for R > 0 sufficiently large. Then there exists an Ro > 0 such that, for everyR > Ro, Ha' is a regular homotopy from H a to H^1 and the morphism Jt
CF*(Ha ) -> CF*(H' ) induced by this homotopy is given by Jt = li3 o i3a.
Proof. This is again a gluing theorem which asserts that for R sufficiently largeone can glue together solutions in M(xa,xi3;Hai3) with those in M(xi3,x';Hi31)
to obtain direct connecting orbits from xa to x' corresponding to the homotopy
H~' · A compactness argument then shows that there are no other solut ions for
large R. Hence there is a bijectionLJM^0 (xa,xi3;Hai3) x M^0 (xi3,x^1 ; Hi3^1 ) ____, M^0 (xa,x^1 ;H~^1 )
x/3
for R sufficiently large. One can show that this bijection is orientation preserving
and this proves the lemma. DLemma 3.12. If Hg^13 and Hf^13 are two regular homotopies from Ha to Hi3, then
the two corresponding chain homomorphisms <I>ga and <I>f a are chain homotopyequivalent. In other words, there exists a homomorphism T: CF(Ha)-> CF(Hi'.1)
such thatf a - ga = ai3T +Toa.
Proof. The idea is to choose a regular homotopy of homotopies H~~,t fromHa to Hi3 which agrees with Hg~ t for .A = 0 and Hf~ t for .A = l.
Then one considers the parametri;ed moduli space M-^1 (;a', yi3; {H~^13 }) =
{ (.A,u): 0 :<:::.A:<::: l,u E M(xa,xi3;H~^13 ),μ(u;H~^13 ) = -1}. This moduli space is a
zero dimensional manifold (for a generic homotopy of homotopies) and the chain
homotopy T: CF(Ha ) -> CF(Hi'.1) is defined by
T(xa) = L #M-l(xa, yi3)(yi3)
yf3where #M-^1 (xa, yi'.1) is to be understood as counting with appropriate signs. Note
that this zero dimensional moduli space consists of finitely many pairs (>- 1 , u 1 ) with
O < .A 1 < l. Since the homotopies Hg^13 and Hf^13 are regular there cannot be
any connecting orbits with index -1 for .A = 0 and .A = l. However, in a generic
1-parameter family such orbits do occur for isolated parameter values. Counting
these gives rise to the chain homotopy equivalence T. That T satisfies the required
equation follows again by examining the boundaries of the 1-dimensional moduli
spaces (see Figure 10). One finds
aM^0 (ya, yi3; { H~^13 }) M^0 (ya, y^13 ; Hg^13 ) U M^0 (ya, y^13 ; Hf^13 )
U LJ M-^1 (ya, zi3; {H~^13 }) x M^1 (zi3, yi3; Hi3)
x"The proof involves again Floer's gluing argument and Gromov compactness. The
identity is to be understood with orientations, and this proves the lemma. D