LECTURE 3. FLOER HOMOLOGY 187

3.8. Floer homology revisited

It is useful to introduce a covering of the space .CM of contractible loops in M

with covering group r. For every contractible loop x: IR/Z -t M choose a smooth

map u : B -t M defined on the unit disc B = {z E <C : lzl ::; 1} which satisfies

u(e^2 ,,.it) = x (t). Two such maps u 1 and u 2 are called equivalent if their sum

u 1 #( -u2) represents a torsion homology class. We use the notation [x, u 1 ] ""

[x, u2] for equivalent pairs and denote by .CM the space of equivalence classes. The

elements of .CM will also be denoted by x. The space .CM is the unique covering

space of .CM whose group of deck transformations is the image r c H 2 (M) of the

Hurewicz homomorphism 7r 2 (M) -t H 2 (M). We denote by

r x .CM -t .CM : (A, x) rt A#x

the obvious action off on .CM. The symplectic action functional aH: .CM -t JR

is defined by

aH([x, u]) = - l u*w - fo

1

Ht(x(t)) dt

and satisfies

aH(A#x) = aH(x ) - w(A)

for A E r. Let us denote by f->(H) C .CM the covering of the set P(H) of con-

tractible periodic solutions of ( 1).
The Floer chain complex can now b e introduced as the set CF*(H) of formal
sums of the form
~ = I: 6(x)
ioE'P(H)
which satisfy the finiteness condition

{ x E f->(H) : ~x i-0, aH(x) ;::: c} < oo

for every constant c ER The Novikov ring Aw obviously acts on CF*(H) by

>. * ~ = L L >.A~x(A#x).

AHxEP(H)

Thus(>.* O x = l:A AA ~(-A)#x and the reader may check that these elements still

satisfy the required finiteness condition. Now one can proceed as before and define
the Floer boundary map

f)F : CF(H) -t CF(H)

by counting the connecting orbits in M(y, x; H, J) in the case of index difference 1.
The pair (ii, x) encodes the homology class of the connecting orbits, and in the case
of index difference 1 this moduli space is a finite set. The resulting Floer boundary
map is then a module homomorphism over the Novikov ring Aw and hence the
Floer homology groups are modules over Aw. The analogue of Theorem 3.7 for

Calabi-Yau manifolds is that the Floer homology groups of Hand J are naturally

isomorphic to the singular homology of M with coefficients in the Novikov ring:

HF(M,w,H,J) ~ H(M;Aw)·

The Arnold conjecture for Calabi-Yau manifolds follows immediately from this as-
sertion. For details see Hofer-Salamon [19].