Poincaré duality IV (Ex)

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(Created page with "<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_*) \righta...")
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Revision as of 10:38, 22 March 2012

Exercise 0.1. Let $C_*$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}C_*$ and $D_*$$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.

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