186 D. SALAMON, FLOER HOMOLOGY
diverging to infinity. In other words, the moduli space M^1 (x, y; H, J) may not be
a finite set but there are finitely many connecting orbits in each homology class.
Counting the connecting orbits in their homology classes leads naturally to Floer
homology over Novikov rings [19].
- Novikov rings
Continue to assume (36) and let r c H2(M) = H 2 (M,Z)/torsion be the image of
the Hurewicz homomorphism 7r 2 (M) ____, H 2 (M). Associated to the homomorphismw : r ____,IR is the Novikov ring A= Aw. This is a kind of completion of the group
ring of r, reminiscent of the ring of Laurent series. The elements of the Novikov
ring A are formal sums of the form
A = L AA e21riA
AEr'
with AA E Z, which satisfy the finiteness condition
#{A Er: AA -1-0, w(A) ~ c} < oofor every c > 0. In other words, for each c > 0, there are only finitely many nonzero
coefficients AA with energy w(A) ~ c. The ring structure is given by
A * μ = L AAμse2.,.-i(A+B).
A,BThus (A* μ)A = 2: 8 AA-BμB. It is a simple matter to check that the finiteness
condition is preserved under this multiplication.
Remark 3.13. (i) The Novikov ring can also be defined if the first Chern
class does not vanish over 7r 2 (M). In that case the Novikov ring carries a
natural grading given by the first Chern class via
deg( e^2 .,.-iA) = 2c1 (A).(ii) If c 1 -1- 0 we denote by Ak C A the subset of all elements of degree k.
Then Ao is a ring, but in general multiplication changes the degree via the
formuladeg(A * μ) = deg(A) +deg(μ).
In other words, Ak is a module over A 0. Moreover, multiplication by anyelement of degree k provides a bijection Ao ____, Ak. Note that Ak -1- 0 if and
only if k is an integer multiple of 2N where N is the minimal Chern numberof M (defined by (c 1 ,7r 2 (M)) =NZ).
(iii) If r = Z then A is the ring of Laurent series with integer coefficients.
This is a principal ideal domain and if the coefficients are taken in a field
then A is a field. These observations remain valid when the homomorphismw: r ____,IR is injective. (See for example [19].) In the case 7r 2 (M) =Zit is
interesting to note the difference between c 1 = 0 and c 1 = [w]. In both cases
Aw is the ring of Laurent series but if c 1 = 0 then this ring is not graded
and in general we cannot exclude the possibility of infinitely many nonzero
coefficients.
(iv) Novikov first introduced a ring of the form Aw in the context of his Morsetheory for closed 1-forms (cf. [ 34]). In that case r is replaced by the funda-
mental group and the homomorphism 7r 1 (M) ____, IR is induced by the closed
1-form. D