Poincaré duality III (Ex)

Latest revision as of 14:56, 1 April 2012

Let $<wikitex>; Let $M$ be # $S^1$ # $S^1\times S^1$ # $\mathbb{R} P^2$ # $K$ the Klein bottle Consider all possible representations $\omega: \pi_1(M) \to \mathbb{Z}_2$. Compute * $H^*_{\mathbb{Z},\omega}(\widetilde{M}):= H^*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(\Hom_{\mathbb{Z}\pi_1(M)}(C(\widetilde{M}),\mathbb{Z}\pi_1(M)_\omega)).$ * $H_*^{\mathbb{Z},\omega}(\widetilde{M}):= H_*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(C(\widetilde{M})\otimes_{\mathbb{Z}\pi_1(M)}\mathbb{Z}\pi_1(M)_\omega).$ 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 $S^1$, why is the correct involution for Poincaré Duality $$ \Sigma {a_jt^j}\mapsto \Sigma a_j \omega (t) t^{-j} $$ and not $$ \Sigma{a_jt^j}\mapsto \Sigma a_j \omega (t) t^j ~? $$ </wikitex> == References == {{#RefList:}} [[Category:Exercises]]M$ be

1. $S^1$
2. $S^1\times S^1$
3. $\mathbb{R} P^2$
4. $K$ the Klein bottle

Consider all possible representations $\omega: \pi_1(M) \to \mathbb{Z}_2$. Compute

• $H^*_{\mathbb{Z},\omega}(\widetilde{M}):= H^*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(\Hom_{\mathbb{Z}\pi_1(M)}(C(\widetilde{M}),\mathbb{Z}\pi_1(M)_\omega)).$
• $H_*^{\mathbb{Z},\omega}(\widetilde{M}):= H_*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(C(\widetilde{M})\otimes_{\mathbb{Z}\pi_1(M)}\mathbb{Z}\pi_1(M)_\omega).$

$$\displaystyle [M]\cap - : \left\{ \begin{array}{c} H^{k}_{\mathbb{Z},\omega}(\widetilde{M}) \to H_{\dim M -k}(\widetilde{M}) \\ H^{k}(\widetilde{M}) \to H^{\mathbb{Z},\omega}_{\dim M -k}(\widetilde{M}) \end{array}\right. ?$$

$$\displaystyle \Sigma {a_jt^j}\mapsto \Sigma a_j \omega (t) t^{-j}$$

$$\displaystyle \Sigma{a_jt^j}\mapsto \Sigma a_j \omega (t) t^j ~?$$