4. Some algebra for Floer theory 89
4.2. Abstract Floer theory
In this section, we present an axiomatic model for Morse and Floer theory over possibly non-compact manifolds. We follow [15], who claim that it all goes back to a suggestion made by D. Salamon. The Main Theorem 81 states that the two basic approaches in this setup - working over a Novikov completion or taking limits - yields the same homology, i.e. that F H ∼= lim
−→lim
←− F H[a,b]. Our proof of this crucial theorem is less abstract (compared to [15], though perhaps tedious) and completely selfcontained.
Definition 76. A Floer triple over a ring R is a tuple F = (C, f, m) consisting of a set C, a function f : C → R and a function m : C ×C → R such that the following conditions hold.
i. The set Cab :={c∈C|a≤f(c)≤b} is finite for each −∞< a≤b <∞.
ii. If m(c1, c2)̸= 0 then f(c1)≤f(c2).
iii. For all c1, c3 ∈C holds
c2∈C
m(c1, c2)·m(c2, c3) = 0.
Remark.
• Assertionsi. and ii. assure that the sum in iii. is actually finite.
• We callf the action function, call the elements of C critical points of action f(c) and callCab the set of critical points in the action window [a, b]. The valuem(c1, c2) represents the count (with signs) of Floer cylinders with asymptotics c1 and c2.
• The condition f(c1) ≤ f(c2) in assertion ii, reflects the fact that we also include Morse-Bott theories. For pure Morse theories replace “≤” by “<”.
• Assertion i. implies that the action spectrum spec(f) := f(C) ⊂R is closed and discrete. We can hence find a monotone injective map from f(C) to Z. With the help of this map we will henceforth assume that in fact
f(C)⊂Z.
This is solely for notational reasons and does not effect the generality of our the-orems. Under this assumption, the first assertion takes the following form
i’. For every a∈Z, the set Caa =f−1(a) = {c∈C|f(c) =a} is finite.
• Note the changed notation for the action window in this chapter. In all other chapters we writeC(a,b) instead of Cab. In particular, the lower index denotes the lower end of the action window and not the grading with respect to the boundary operator, which plays only a minor role in this chapter.
• Assertion iii. guarantees that the formula dc0 :=
c∈Cm(c, c0)·c will define a boundary operator.
Definition 77. For a ≤ b ∈Z let F Cab :=Cab⊗R be the free R-module generated by Cab. For anya ≤c≤b let
πcab :F Cab −→F Ccb ∼=F Cab
F Cac−1 and ιbca :F Cac−→F Cab ∼=F Cac⊕F Cc+1b be the natural projection resp. inclusion.
Obviously, (F Cab, π, ι) is a bidirect system overZ×Z, where the quasi order in the upper index is “≤”, while it is “≥” on the lower one.
Definition 78. We set F Cb := lim
←−a
F Cab and F C := lim
−→
b
F Cb = lim
−→
b
lim←−
a
F Cab. Note thats∈F Cb can be interpreted as a formal sum s =
a≤b
sa, sa∈F Caa. Then,s ∈F C is also interpreted as a formal sum s=
a∈Z
sa, sa∈F Caa.
As F C is obtained by a direct sum, we find for the second sum that there exists a b ∈ Z such that sa = 0 for all a > b. Therefore, we write the elements of F C also as s=
a≤bsa. Note that the mapsπ andι extend to the limits and we have in particular πcb : F Cb →F Ccb,
a≤b
sa→→
b
a=c
sa.
We call the summands sa in the representation of s ∈ F C the a-th coefficient of s and we use for theA-th coefficient of s the notation
KA(s) = KA
a≤b
sa
:=sA.
Note thatKa :F C →F Caaas a map is well-defined and linear. The following important fact is an easy consequence of the definition ofF C.
Lemma 79. Any sequence of coefficients (sa)a∈Z with sa = 0 for all a > b with some b∈Z determines uniquely an element in F C.
Now let us take a strictly decreasing sequence (sb)b≤b0 ⊂F C, sb =
a≤bsba ∈F Cb. We associate to the formal Laurent series
b≤b0sb a value inF C by the well-known formula
b≤b0
sb :=
a≤b0
b0
b=a
sba=
a≤b0
b0
b=a
Ka(sb).
Observe that the inner sum is finite thus determining an element sa ∈F Caa. The value of the infinite sum
sb is then the unique element in F C represented by the sequence of coefficients (sa)a≤b0. The well-known features of this infinite sum are summarized in the following lemma.
Lemma 80. For b0 ∈Z and b≤b0 let sb, tb ∈F Cb. Then
d(sb). (countable linearity) Proof: We calculate
So all coefficients of the left side of equation (a) coincide with all coefficients of the right side. Hence they represent the same element inF C. The proof of the subsequent assertions follows the same scheme.
(b) KA
On the other hand KA
b≤b0
d(sb)
=KA
a≤b0
b0
b=a
Ka
d(sb)
=
b0
b=A
KA d(sb)
. We define a boundary operator d onF Cab as the linear extension of
dx:=
y∈Cab
m(y, x)·y, x∈Cab.
Note that assertion iii. implies thatd2 = 0, i.e. thatdreally is a boundary operator. As d does not increase action (due to ii.), we find thatd satisfies
πcab ◦d=d◦πcab and ιbca ◦d=d◦ιbca ∀a≤c≤b. (42) Therefore,dinduces maps onF Cb andF C, which we will also denote withd. Explicitly, they are given as the infinite linear extension of
dx:=
y∈C
m(y, x)·y, x∈C.
Note that ii. and iii. imply thatd2 = 0 and that (42) still holds, now including a=−∞
and b =∞. Note that we may express the boundary operator onF Cab as πba◦d, where d is the operator on F C. We denote the resulting homologies by
F Hab, F Hb and F H.
The elements of these homologies are denoted by [s]ba,[s]b and [s], where we suppress the upper index whenever it is clear from the context.
As a consequence of (42), we find thatπ and ιdescend to linear maps in homology, still denoted byπ and ι:
πbca : F Hab →F Hcb ιbca : F Hac→F Hab ∀ − ∞ ≤a≤c≤b ≤ ∞.
As (F Cab, π, ι) resp. (F Cb, ι) are bidirect resp. direct systems, it follows that (F Hab, π, ι) resp. (F Hb, ι) are bidirect resp. direct systems as well. Observe that we have in particular for every fixedb0 and any a≤b0 the following map
πab0 : F Hb0 →F Hab0, [s]→→b0
b=a
sb
a
, where s =
b≤b0
sb ∈F Cb0,
which satisfiesπbc0 =πcab0πab0 for everya≤c. Hence, there exists by the universal property a uniqueR-linear map
ϕ : F Hb0 →lim
←−a
F Hab0, [s]→→
b0
b=a
sb
a
a≤b0
.
Analogously, we have for everyb ∈Za map
a) The map ϕ is always surjective. If R is a field, it is injective and yields for every b∈Z an isomorphism
F Hb ∼= lim
In the notation of the other sections, this reads as F H ∼= lim
−→
(a) claim: ϕ is surjective.
proof: Let (σa)a≤b ∈ lim
For the construction of thesa first consider the following short exact sequence 0→F Caa −→ι F Cab −→π F Ca+1b →0,
which yields in homology the long exact sequence
· · · →F Ha+1b −→δ F Haa −→ι F Hab −→π F Ha+1b −→δ F Haa →. . . (43) Here, the connecting homomorphismδ is given by
δ[s]ba+1 := πadsa
a,
which is the class of thea-th coefficient of ds, as ds ∈F Ca.
Here and in the following we suppress the indices and writeι orπ instead ofιbaa or πa+1,ab , whenever it is clear from the context.
Let sb ∈ F Cbb be any representative of σb, i.e. σb = [sb]b. Now assume that claim: ϕ is injective if R is a field.
proof: Assume that ϕ [s] For the empty sequence, condition (∗∗) has to hold forA =b, which is true by (∗).
Now assume thatnb, . . . , nA+1 have already been constructed such that (∗∗) holds. (∗∗), we find that Gk ̸= ∅. Moreover, Gk has the structure of an affine subspace ofF CAA and Gk+1 ⊂Gk. As F CAA is a finite dimensional vector space (R is a field and CAA finite), we find that the common intersection is not empty, i.e.
G:=
k≥0
Gk ̸=∅.
Choose any nA∈G. The construction of Gshows that (∗∗) holds fornb, . . . , nA.
(b) claim: ψ is injective.
proof: Assume that ψ n
i=1σi
= 0 ∈ F H for some n i=1σi
∈ lim
−→ F Hb, σi ∈F Hbi. Letsi ∈F Cbi be such that σi = [si]bi. It follows thatψ n
i=1σi
=
n i=1si
= 0.Hence, there exists a β ∈F C such that n
i=1si =dβ. Then there exists a b0 ∈Z such that b0 ≥bi for all iand β ∈F Cb0 ⊂F C. This implies that
n
i=1
ιb0biσi =n
i=1
sib0
= dβb0
= 0∈F Hb0, which shows that n
i=1σi
= 0∈lim
−→ F Hb. claim: ψ is surjective.
proof: Consider any [s] ∈ F H, s ∈ F C. Then there exists a b0 ∈ Z such that s∈ F Cb0. As ds = 0, we find that s represents a well-defined class [s]b0 ∈F Hb0. Let
[s]b0
denote its class in lim
−→ F Hb. Then we see that ψ
[s]b0
= [s], thus showing thatψ is surjective.