Poincaré duality IV (Ex)

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

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

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

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

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

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