Poincaré duality IV (Ex)

Exercise 0.1. Let $<wikitex>; {{beginthm|Exercise}} \label{chain_map} Let $ C_* $ and $ D_* $ be $ \Zz\pi $-chain complexes and $$ s: \Zz^{\omega} \otimes_{\Zz\pi}(C_* \otimes_{\Zz} D_*) \rightarrow Hom_{\Zz\pi}(C^{-*},D_*) $$ 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 $$ s(1 \otimes x \otimes y): (C^{-*})_{k-n} \rightarrow D_k, (\phi:C_{n-k} \rightarrow \Zz \pi) \mapsto \overline{\phi(x)}\cdot y $$ Show that $s$ is a $ \Zz $-chain map. {{endthm}} '''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 $d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$. </wikitex> The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling. == References == {{#RefList:}} [[Category:Exercises]]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.

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

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

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

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.