The Duality Theorem
Poincar´e Duality Section 3.3 239 The Duality Theorem The form of Poincar´e duality we will prove asserts that for an R orientable closed k → n manifold, a certain naturally defined map H (M; R) Hn−k(M; R) is an isomor- phism. The definition of this map will be in terms of a more general construction called cap product, which has close connections with cup product. Cap Product Let us see how cup product leads naturally to cap product. For an arbitrary space X and coefficient ring R let us for simplicity write Ck(X; R) as Ck and its k = k ∈ k ∈ ` dual C (X; R) HomR(Ck,R) as C . For cochains ϕ C and ψ C we have their + cup product ϕ ` ψ ∈ Ck ` . When we regard ϕ ` ψ as a function of ψ for fixed ϕ, + + we get a map C`→Ck ` , ψ , ϕ ` ψ. One might ask whether this map C`→Ck ` is → the dual of a map Ck+` C` . The answer turns out to be yes, and this map on chains will be the cap product map α , α a ϕ. In order to define α a ϕ it suffices by linearity to take α to be a singular simplex + σ : ∆k `→X , and then the formula for σ a ϕ is rather like the formula defining cup product: a = | ··· | ··· σ ϕ ϕ σ [v0, ,vk] σ [vk, ,vk+`] To say that for fixed ϕ the map α , α a ϕ has as its dual the map ψ , ϕ ` ψ is --------------a→ϕ -----→ψ ` to say that the composition Ck+` C ` R is the map α , (ϕ ψ)(α),orin other words, (∗)ψ(αaϕ) = (ϕ ` ψ)(α) + This holds since for σ : ∆k `→X we have | | ψ(σ a ϕ) = ψ ϕ σ|[v , ···,v ] σ|[v , ···,v + ] 0 k k k ` = | ··· | ··· = ` ϕ σ [v0, ,vk] ψ σ [vk, ,vk+`] (ϕ ψ)(σ ) Another way to write the formula (∗) that may be more suggestive is in the form hα a ϕ, ψi=hα, ϕ ` ψi h− −i × i→ where , is the map Ci C R evaluating a cochain on a chain.
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