Poincaré duality III (Ex)
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Patrickorson (Talk | contribs) (Created page with "<wikitex>; Let $M$ be # $S^1$ # $S^1\times S^1$ # $\mathbb{R} P^2$ # $K$ the Klein bottle Consider all possible representations $\omega: \pi_1(M) \to \mathbb{Z}_2$. Comput...") |
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* $H_*^{\mathbb{Z},\omega}(\widetilde{M}):= H_*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(C(\widetilde{M})\otimes_{\mathbb{Z}\pi_1(M)}\mathbb{Z}\pi_1(M)_\omega).$ | * $H_*^{\mathbb{Z},\omega}(\widetilde{M}):= H_*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(C(\widetilde{M})\otimes_{\mathbb{Z}\pi_1(M)}\mathbb{Z}\pi_1(M)_\omega).$ | ||
− | For what $\omega$ do we get Poincaré Duality $$ [M]\cap - : \left\{ \begin{array}{c} H^{k}_{\Z,\omega}(\widetilde{M}) \to H_{\dim M -k}(\widetilde{M}) \\ H^{k}(\widetilde{M}) \to H^{\Z,\omega}_{\dim M -k}(\widetilde{M}) \end{array}\right. ? $$ | + | For what $\omega$ do we get Poincaré Duality $$ [M]\cap - : \left\{ \begin{array}{c} H^{k}_{\mathbb{Z},\omega}(\widetilde{M}) \to H_{\dim M -k}(\widetilde{M}) \\ H^{k}(\widetilde{M}) \to H^{\mathbb{Z},\omega}_{\dim M -k}(\widetilde{M}) \end{array}\right. ? $$ |
For $S^1$, why is the correct involution for Poincaré Duality $$ \Sigma {a_jt^j}\mapsto \Sigma a_j \omega (t) t^{-j} $$ and not $$ \Sigma{a_jt^j}\mapsto \Sigma a_j \omega (t) t^j ~? $$ | For $S^1$, why is the correct involution for Poincaré Duality $$ \Sigma {a_jt^j}\mapsto \Sigma a_j \omega (t) t^{-j} $$ and not $$ \Sigma{a_jt^j}\mapsto \Sigma a_j \omega (t) t^j ~? $$ | ||
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Revision as of 21:45, 19 March 2012
Let be
- the Klein bottle
Consider all possible representations . Compute
For , why is the correct involution for Poincaré Duality
and not