Poincaré duality III (Ex)

Let $M$$\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}{^}M$ be

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

Consider all possible representations $\omega: \pi_1(M) \to \mathbb{Z}_2$$\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)).$$H^*_{\mathbb{Z},\omega}(\widetilde{M}):= H^*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(\Hom_{\mathbb{Z}\pi_1(M)}(C(\widetilde{M}),\mathbb{Z}\pi_1(M)_\omega)).$
• $H_*^{\mathbb{Z},\omega}(\widetilde{M}):= H_*(M; \mathbb{Z}\pi_1(M)_\omega)) = H_*(C(\widetilde{M})\otimes_{\mathbb{Z}\pi_1(M)}\mathbb{Z}\pi_1(M)_\omega).$$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$$\omega$ do we get Poincaré Duality

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

For $S^1$$S^1$, why is the correct involution for Poincaré Duality

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

and not

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