Reducing filtered complexes

 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 ≥b_i for all iand β ∈F C^b0 ⊂F C. This implies that

n



i=1

ι^b0biσⁱ =ⁿ

i=1

sⁱb0

= dβb0

= 0∈F H^b0, which shows that n

i=1σⁱ

= 0∈lim

−→ F H^b. claim: ψ is surjective.

proof: Consider any [s] ∈ F H, s ∈ F C. Then there exists a b0 ∈ Z such that s∈ F C^b0. As ds = 0, we find that s represents a well-defined class [s]^b0 ∈F H^b0. Let 

[s]^b0

denote its class in lim

−→ F H^b. Then we see that ψ

[s]^b0

= [s], thus showing thatψ is surjective.

4.3. Reducing filtered complexes

Recall from the previous section that abstract Floer theory is based on a set C which generates (countably linear) the chain complexF C and the homology F H. Now we are going to address the question to what extend we can reduce C and still get the same homology F H. Under the assumption that R is a field such that all F H_a^b are vector spaces, we will show that we can replace for anya the setC_a^a by any basis of F H_a^a. The motivation behind this is the following. Given a Morse-Bott setup,C_a^a is the set of all critical points of a Morse function on the critical manifoldN^a on the action levela, while the groupsF H_a^a are the Morse homology of N^a. The fact that we can built F H fromF H_a^a then shows that for calculations of F H we may always algebraically pretend that we have a perfect Morse function on N^a, as the critical points of such a function form a basis ofF H_a^a. This will be used in Section 7.2 when we calculate the Rabinowitz-Floer homology for some Brieskorn manifolds, where we know the singular homology of the critical manifolds, but we do not know if we have perfect Morse functions.

To be more explicit, let the reduced chain groups be FC := lim

−→

b

lim←−

a

b

c=aF H_c^c

and consider the following variation of the long exact sequence (43):

· · · →F H_b^b −→^δ F H^b−1 −→^ι F H^b −→^π F H_b^b −→^δ F H^b−1 →. . . , (44) where the map δ is induced by the boundary operatord and given by δ[n_b]^b_b = [dn_b]^b−1. We want to use δ to define a boundary operator on FC. For that, let

K :=δ(FC)⊂lim

−→ lim

←−

b

c=aF H^c−1.

Using the sequence (44), we define below a (non-canonical) map Φ :K →FC. Setting

∂ := Φ◦δ, we will see that ∂² = 0. The Reduction Theorem 85 then tells us that the homology of 

FC, ∂

is isomorphic to F H. Here, the isomorphism is given by a map Ψ : kerδ → F H. We will use this map to show in Theorem 82 that always (not only over field coefficients)FC generatesF H, i.e that the whole homology cannot have more elements then all the singular homologies of the critical manifolds togehter.

Note that one could take Φ = π. However, the resulting homology (FC, ∂) is then isomorphic to the second page of the spectral sequence of the filtration, which is in general not isomorphic toF H. The reason for this discrepancy is thatπ discards kerπ =im(ι).

Our Φ will also use the (kerπ)-part of K.

Remark. Let Ca^a denote a basis of F H_a^a and set C = Ca^a. Then we find that C generates FC. The isomorphism between (FC, ∂) and F H then shows that we can replace C by C and obtain the same homology F H. Moreover, as δ decreases the action, so does∂. Thus, we reduce the Morse-Bott situation to a Morse situation.

To start, we note that the maps in (44) are explicitly given by ι In the following, we are interested in bounded infinite sums

b≤b0η_b of classes η^b ∈ ℵ^b, i.e. in elements of the space

ℵ:= lim Note that in generalρ is neither unique nor does there exist anyR-linearρ ! However, one can pick an R-linear ρ if the sequence splits at π, in particular if R is a field or semi-simple. Now we define Ψ by setting

Ψ

b≤b₀s^b is a cycle, whose class does not depend on the particular choice of the representatives^b for ρ(η^b). Indeed we have:

d anR-homomorphism! However, ifρ isR-linear, then Ψ becomes R-linear as well.

Theorem 82 (Representation Theorem).

The homology F H is represented by ℵ that is Ψ(ℵ) =F H, i.e. Ψ is surjective.

Remark. It follows from this theorem that F H as a set cannot be larger then ℵ. Note that if we have a grading of the complex, then this theorem and all following extend to the graded homology groups and reads as Ψ(ℵ_k) = F H_k.

In order to obtain more precise information, we are now going to analyze the “kernel”

of Ψ, i.e. the preimage Ψ⁻¹(0). To this purpose define K^b :=im(δ) =δ whereK is the space of bounded infinite sums 

k≤b0κ^b, κ^b ∈K^b. Again,j is in general only a map, but can be chosen R-linear if the sequence splits atι.

Then, for By the construction of Φ, it then follows that Φ 

b≤b0κ^b

For anyB ≤b₀ we calculate may without loss of generality assume that s^b0 =dn^b0+1. It also follows from the definition of Φ that

. Repeating the same arguments iteratively then shows thatκ^b = 0 for all b≤b₀ and therefore that

b≤b₀κ^b = 0.

We defined above FC := lim

−→lim

←−

b

c=aF H_c^c. Note that ℵ= kerδ is a natural subset of FC. The limit of the maps δ:F H_b+1^b+1 →im(δ)⊂F H^b is an R-linear mapδ :FC →K.

IfR is a field or semi-simple, we can choose Φ to be R-linear. As mentioned above, we then define an R-linear operator ∂ onFC by

∂ := Φ◦δ.

Sinceim(Φ)⊂ ℵ= ker(δ), we find that ∂² = 0, i.e. that ∂ is a boundary operator.

Theorem 85 (Reduction Theorem).

Let R be a field (or semi-simple). Then, Ψ induces a filtration preserving isomorphism between the homology of (FC, ∂) and F H.

As FC is generated by F H_a^a, a ∈ Z, we can hence in a Morse-Bott setup always alge-braically pretend thatF H is built from the singular homologies of the critical manifolds.

Proof: AsRis a field (or semi-simple), we may choose Φ and Ψ to beR-linear. Then it follows from Lemma 84 that ker(Φ) = Φ⁻¹(0) ={0}. This shows that Φ is injective and hence ker(∂) = ker(Φ◦δ) = ker(δ) =ℵ. Since im(∂) = Φ(δ(FC)) = Φ(im(δ)) = Φ(K), we have for the homology of (FC, ∂) that

ker(∂)

im(∂) =ℵ

Φ(K).

By Lemma 83, we have Φ(K) = ker(Ψ). Hence, Ψ induces a well-defined injective map Ψ : ker(∂)

im(∂) =ℵ

Φ(K) −→F H, (∗)

which we denote by abuse of notation also Ψ. As Ψ is by Theorem 82 surjective, we find that (∗) is in fact an isomorphism. The action filtration preserving property is obvious by the construction of Ψ (see (46)).

Examples. Let us consider 4 critical pointsC ={a₁, b₁, a₀, b₀}, where the index denotes the action, i.e. f(a_k) =k. We set

m(b₁, a₁) = 2, m(b₀, a₁) = 1 and m(b₀, a₀) = 2.

Then da₁ = 2b₁+b₀, da₀ = 2b₀ db₁ =db₀ = 0.

In a graded context, one should think of thea_k as having one index higher then the b_k. UsingZ-coefficients, we get

F H₁¹ =⟨b₁⟩

⟨2b₁⟩ ∼=Z2 F H₀⁰ =F H⁰ =⟨b₀⟩

⟨2b₀⟩ ∼=Z2

F H =F H₀¹ =⟨b₀, b₁⟩

⟨2b₀,2b₁+b₀⟩ = ⟨2b₁+b₀, b₁⟩

⟨−4b₁,2b₁+b₀⟩ ∼=Z⁴. This shows that there is no hope of building a chain complex fromF H₁¹ andF H₀⁰ whose homology is F H, if we do not use field coefficients.

If we takeZ2-coefficients instead, we get da₁ =b₀ and da₀ =db₁ =db₀ = 0. Then F H₁¹ =⟨a₁, b₁⟩ ∼=

Z2

2

, F H₀⁰ =F H⁰ =⟨a₀, b₀⟩ ∼= Z2

2

, F H =F H₀¹ =⟨a₀, b₀, b₁⟩

⟨b₀⟩ ∼= Z2

2

. The only relevant sequence here is

F H⁰ −→^ι F H¹ −→^π F H₁¹ −→^δ F H⁰,

where δ maps the class of b₁ to 0 and the class of a₁ to the class of b₀, as da₁ = b₀ and db₁ = 0. As F H⁰ = F H₀⁰ no further maps have to be constructed. The boundary operator ∂ on the complex FC = 

[a1],[a0],[b1],[b0]

is thus given by ∂[a1] = [b0] and

∂[a₀] =∂[b₁] =∂[b₀] = 0. The homology of this complex is therefore

FC, ∂

=

[a₀],[b₀],[b₁] 

[b₀] ∼= Z2

2

, which is trivially isomorphic toF H.

Im Dokument Rabinowitz-Floer homology on Brieskorn manifolds (Seite 107-115)