# Poincaré duality (Ex)

Let $\Zz_\omega$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Zz_\omega$ the local coefficient system of $\Zz$$\Zz$ twisted the orientation character $\omega \colon \pi_1(M) \to \Zz/2$$\omega \colon \pi_1(M) \to \Zz/2$ of a compact manifold $M$$M$, let $S^{n-1} \tilde \times S^1$$S^{n-1} \tilde \times S^1$ denote the total space of the non-trivial linear sphere bundle over $S^1$$S^1$ and let $N$$N$ be a closed simply connected manifold.

Determine the following homology groups (in terms of $H_*(N; \Zz$$H_*(N; \Zz$ where appropriate):

1. $H_*(\RP^n; \Zz)$$H_*(\RP^n; \Zz)$
2. $H_*(\RP^n; \Zz_\omega)$$H_*(\RP^n; \Zz_\omega)$
3. $H_*(S^{n-1} \tilde \times S^1; \Zz)$$H_*(S^{n-1} \tilde \times S^1; \Zz)$
4. $H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$$H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$
5. $H_*(\RP^n \sharp N; \Zz_\omega)$$H_*(\RP^n \sharp N; \Zz_\omega)$
6. $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$$H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$
7. $H_*(S^1 \times N; \Zz_\omega)$$H_*(S^1 \times N; \Zz_\omega)$
8. $H_*(\RP^n \times N; \Zz_\omega)$$H_*(\RP^n \times N; \Zz_\omega)$

Determine the following cohomology groups and verify Poincaré duality using the homology computations above:

1. $H^*(\RP^n; \Zz_\omega)$$H^*(\RP^n; \Zz_\omega)$
2. $H^*(\RP^n; \Zz)$$H^*(\RP^n; \Zz)$
3. $H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)$$H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)$
4. $H^*(S^{n-1} \tilde \times S^1; \Zz)$$H^*(S^{n-1} \tilde \times S^1; \Zz)$
5. $H^*(\RP^n \sharp N; \Zz)$$H^*(\RP^n \sharp N; \Zz)$
6. $H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)$$H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)$
7. $H^*(S^1 \times N; \Zz)$$H^*(S^1 \times N; \Zz)$
8. $H^*(\RP^n \times N; \Zz)$$H^*(\RP^n \times N; \Zz)$

Here are some helpful references for the definitions involved: [Davis&Kirk2001, Ch 5] [Wall1967a, Chapter 1]