9. Strong Lefschetz Fixpoint Theorem. 1. Products.

9. Strong Lefschetz Fixpoint Theorem. 1. Products. For singular cohomology of a space X we have the following products: Cup Product. ∪ : H k (X) × H ℓ (X) → H k+ℓ (X), ([ϕ], [ψ]) 7→ [ϕ ∪ ψ] where (ϕ ∪ ψ)(σ) := ϕσ[v0 , . . . , vk ] · ψσ[vk , . . . , vk+ℓ ]. Cross Product. × : H k (X) × H ℓ (Y ) → H k+ℓ (X × Y ), (ϕ, ψ) 7→ ϕ × ψ, where ϕ × ψ := p∗1 (ϕ) ∪ p∗2 (ψ) Here p1 : X × Y and p2 : X × Y are the projections onto the two factors.

Klaus Johannson, Homology of Chain Complexes

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. Strong Fixpoint Theorem

Cap Product. ∩ : Hk X → H ℓ X → Hk−ℓ X, (σ, ϕ) 7→ σ ∩ ϕ, where (σ ∩ ϕ) := ϕ|σ[v0 , . . . , vℓ ] σ|[vℓ , . . . , vk ]. Here σ = [v0 , . . . , vk ] and k ≥ ℓ. Remark. Cup and cross product are related by the formula ϕ ∪ ψ = d∗ (ϕ × ψ), where d : X → X × X, x 7→ (x, x), is the diagonal map. Cup and cap product are related by the formula (ϕ ∪ ψ)(σ) = ψ(σ) ∩ ϕ


§9 Strong Fixpoint Theorem

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2. The Poincare Map. Let X be a space with Hn X ∼ = Z and Hi X = 0, for all i < 0 and i > n (e.g. if X is an n-manifold). A cycle z ∈ Cn X is called a fundamental cycle if [z] generates Hn X ∼ = Z. The map θ : Hk X → H n−k X, ϕ 7→ θϕ , where θϕ (σ) = σ ∩ ϕ is called the Poincare map. Theorem. Let X be a topological n-manifold. Then the Poincare map θ : Hk X → H n−k X given above is an isomorphism. ♦ Let X be a topological manifold of even dimension n = 2q. Then we have a bilinear map (i.e. a pairing) (−.−) : Hq X × Hq X → Z given by (σ, τ ) 7→ σ ∩ θ(τ ). Klaus Johannson, Homology of Chain Complexes

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This pairing is called the algebraic interstion number of σ and τ . Definition. Let f : X → X be a map. Then the integer (Γf , ∆) ∈ Z is called the fixpoint number of f . Remark. Here Γf :={ (x, f (x)) ∈ X × X | x ∈ X } is the graph of f and ∆ :={ (x, x) ∈ X × X | x ∈ X } is the diagonal in X × X.



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3. The Strong Lefschetz Fixpoint Theorem. As we have seen the weak Lefschetz fixpoint theorem predicts the mere existence of a fixpoint (in favourable circumstances) but says nothing about the actual number of fixpoints. It can be proved for simplicial complexes, i.e. for rather general spaces. It turns out that in case of manifolds one can also say something about the number of fixpoints. This is the strong Lefschetz theorem. Actually, one need not restrict to manifolds. More important are the properties of the homology theory. The homology should satisfy the K¨ unneth formula and Poincare duality. The K¨ unneth formula holds for chain complexes and it turns out that it holds for the homology of manifolds. In order to formulate and prove the strong Lefschetz theorem we need some preparation. (1) The algebraic intersection number (x, y) is given by the cup product in cohomology. (2) Let ϕ : X → X be a map. Then the number of fixed points (with multiplicities) of ϕ is given by (Γϕ , ∆), where Γϕ := { (x, ϕ(x)) ∈ X × X | x ∈ X }. Klaus Johannson, Homology of Chain Complexes

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(3) Assuming that we take homology with coefficients in a field k and assuming homology satisfies the (1) K¨ unneth formula and (2) Poincare duality, we want to prove (Γϕ , ∆) =

X

trace(ϕ∗ |H r (X, k)).

r

Let d = dim(X). Then from Poincare duality we have a non-degenerate pairing H ∗ (X, k) × H ∗ (X, k) → H 2d (X, k) ∼ =k Let e2d be the canonical generator of H 2d (X, k). It is the class of any point. In the next lemma p∗ denotes the Gysin map. Lemma 1. Let ϕ : X → Y b ∈ H ∗ Y . Then

be a map between manifolds and

ϕ∗ (y) = p∗ (Γϕ ∪ q ∗ (y)), where

q

p

Y ←− X × Y −→ X.



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Proof. Note that (1, ϕ) : Γϕ ֒→ X × Y . Hence, p∗ (Γϕ ∪ q ∗ y) = p∗ ((1, ϕ)∗ (1) ∪ q ∗ y) = p∗ (1, ϕ)∗ (1 ∪ (1, ϕ)∗ q ∗ y) = (p ◦ (1, ϕ))∗ (1 ∪ (q ◦ (1, ϕ))∗ y) = id∗ (1X ∪ ϕ∗ y) = ϕ∗ (y). ♦ Lemma 2. Let {ei } be a basis of H ∗ (X) and {fi } be the dual basis of H ∗ (X) determined by the Poincare duality map, i.e., ei ∪ fi = δij e2d . Then, for a map ϕ : X → X, X Γϕ = ϕ∗ (ei ) ⊗ fi . Proof. Since fi form a basis of k-vector space H ∗ X, they form a basis for H ∗ (X × X) = H ∗ X ⊗k H ∗ X as a free H ∗ X-module (K¨ unneth formula). Hence Γϕ =

X

ai ⊗ fi ,

i


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for some unique ai ∈ H ∗ X. Then, by lemma above, ϕ∗ (ej ) = p∗ (Γϕ ∪ q ∗ ej ) X = p∗ (( ai ⊗ fi ) ∪ (1 ⊗ ej )) i

= p∗ (aj ⊗ e2d ) = aj . ♦ Theorem. Let X be an object (e.g. a manifold). Suppose the homology of X satisfies the K¨ unneth formula and Poincare duality. Let ϕ : X → X be a map. Then (Γϕ , ∆) =

X

trace(ϕ∗ |H r X).

r

Proof. Let eri be a basis for H r , for H 2d−r .

fi2d−r be the dual basis

Then by the lemma 2, Γϕ =

X

ϕ∗ (eri ) ⊗ fi2d−r

i,r

and



∆ = Γid =

X

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eri ⊗ fi2d−r

i,r

=

X

(−1)r(2d−r) fi2d−r ⊗ eri

X

(−1)r fi2d−r ⊗ eri .

i,r

=

i,r

Hence (Γϕ , ∆) = Γϕ ∪ ∆     X X 2d−r ∗ r ∪ (−1)r fi2d−r ⊗ eri  = ϕ (ei ) ⊗ fi i,r

i,r

=

X

(−1)r ϕ∗ (eri )fi2d−r ⊗ e2d i .

i,r

Write ϕ∗ : H r X → H r X in terms of the basis {eri } as X ∗ r aji,r erj . ϕ (ei ) = j


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Then X

(−1)r ϕ∗ (eri )fi2d−r ⊗ e2d

i,r

=

XX i,r

X

aji,r erj fi2d−r ⊗ e2d

j

(−1)r aii,r e2d ⊗ e2d

i,r

=

X

(−1)r trace(ϕ∗ |H r X)e2d ⊗ e2d .

r

Hence, applying ηX×X on both sides, we obtain (Γϕ , ∆) =

X

(−1)r trace(ϕ∗ |H r X).

r

which is the desired equation. ♦ Remark. In certain situations there may further be a criterion for when (Γϕ , ∆)p = 1 for a fixed point p of ϕ. The proof of the strong Lefschetz fixpoint theorem is taken from [Milne, Lectures on Etale Cohomology, p. 142pp].
