Poincaré duality IV (Ex)
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− | '''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 $d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$. | + | '''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 $d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$. |
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Revision as of 22:37, 22 March 2012
Exercise 0.1. Let and be -chain complexes and
be defined by sending to the map
Show that is a -chain map.
Hint: Note that with boundary map and that with boundary map .
The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.