Poincaré duality III (Ex)

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Let M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_DPijOE 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).
For what \omega do we get Poincaré Duality
\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. ?
For S^1, why is the correct involution for Poincaré Duality
\displaystyle  \Sigma {a_jt^j}\mapsto \Sigma a_j \omega (t) t^{-j}
and not
\displaystyle  \Sigma{a_jt^j}\mapsto \Sigma a_j \omega (t) t^j ~?
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