Poincaré duality IV (Ex)

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Exercise 0.1. Let C_* and D_* be \Zz\pi-chain complexes and

\displaystyle  s: \Zz^{\omega} \otimes_{\Zz\pi}(C_* \otimes_{\Zz} D_*) \rightarrow Hom_{\Zz\pi}(C^{-*},D_*)

be defined by sending 1 \otimes x \otimes y \in \Zz^{\omega} \otimes_{\Zz \pi}(C_{n-k} \otimes_{\Zz} D_k) to the map

\displaystyle  s(1 \otimes x \otimes y): (C^{-*})_{k-n} \rightarrow D_k, (\phi:C_{n-k} \rightarrow \Zz \pi) \mapsto \overline{\phi(x)}\cdot y

Show that s is a \Zz-chain map.

Hint: Note that \Zz^{\omega} \otimes_{\Zz \pi}(C_* \otimes_{\Zz} D_*)_n=\prod_{k \in \Zz} \Zz^{\omega} \otimes_{\Zz\pi}(C_{n-k} \otimes_{\Zz} D_k) with boundary map d=id \otimes ((-1)^kd_C \otimes id + id \otimes d_D) and that Hom_{\Zz\pi}(C^{-*},D_*)_n=\prod_{k \in \Zz}Hom_{\Zz\pi}((C^{-*})_{k-n},D_k) with boundary map d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}.

Exercise 0.2. Let C_* and D_* be \Zz\pi-chain complexes. Show that for the n-th homology of the complex hom_{\Zz\pi}(C_*,D_*) we have

\displaystyle  H_n(hom_{\Zz\pi}(C_*,D_*))= [ \Sigma^n C_*,D_*]_{\Zz\pi}

where (\sum^nC_*)_k:=C_{k-n} denotes the shifted complex with boundary map d_{\sum C}:=(-1)^nd_C.

Hint: use the boundary map of Exercise 0.1.

Exercise 0.3. Deduce from the Poincare homotopy equivalence ? \cap [X]:C^{n-*}(\widetilde X) \rightarrow C_*(\widetilde X) that

\displaystyle  H_k(X,\Zz^{\omega}) \cong H^{n-k}(X,\Zz):=H_k(C^{n-*}(X))

as \Zz-Modules.

Exercise 0.4. Let (X,A) be a Poincare pair with X an n-dimensional, connected, finite CW-complex and A\subset X an (n-1)-dimensional subcomplex, an orientation homomorphism \omega_X : \pi \longrightarrow \{\pm 1\} and a fundamental class [X,A]\in H_n(X,A;\Zz^{\omega_X}):=H_n(\Zz^{\omega_X}\otimes_{\Zz\pi}C_*(\tilde{X},\tilde{A})). That means, for a universal covering p:\tilde{X}\longrightarrow X and \tilde{A}:=p^{-1}(A) the \Zz\pi-chain maps ?\cap [X,A]:C^{n-*}(\tilde{X},\tilde{A})\longrightarrow C_*(\tilde{X}) and ?\cap [X,A]:C^{n-*}(\tilde{X})\longrightarrow C_*(\tilde{X},\tilde{A}) are \Zz\pi-chain homotopy equivalences.

Show that the components C\in\pi_0(A) of A inherit the structure of a finite (n-1)-dimensional Poincare complex, i.e. that there is an induced orientation homomorphism \omega_C : \pi_1(C) \longrightarrow \{\pm 1\} and an induced fundamental class [C]\in H_{n-1}(C;\Zz^{\omega_C}), such that

\displaystyle  ?\cap [C]:C^{n-1-*}(\tilde{C})\longrightarrow C_*(\tilde{C})

is a \Zz\pi-chain homotopy equivalence.

Hint: Tensorize both sides with \Zz^{\omega} and consider the induced map.

The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.

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