Poincaré duality (Ex)

Let $\Zz_\omega$$<wikitex>; Let $\Zz_\omega$ denote homology with local co-efficients in $\Zz$ twisted the orientation character $\omega \colon \pi_1(M) \to \Zz/2$ of a compact manifold $M$, let $S^{n-1} \tilde \times S^1$ denote the total space of the non-trivial linear sphere bundle over $S^1$ and let $N$ be a closed simply connected manifold. Determine the following homology groups (in terms of $H_*(N; \Zz)$ where appropriate * $H_*(\RP^n; \Zz)$ * $H_*(\RP^n; \Zz_\omega)$ * $H_*(S^{n-1} \tilde \times S^1; \Zz)$ * $H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$ * $H_*(\RP^n \sharp N; \Zz_\omega)$ * $H_*(*S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ </wikitex> == References == {{#RefList:}} [[Category:Exercises]]\Zz_\omega$ denote homology with local co-efficients in $\Zz$$\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

• $H_*(\RP^n; \Zz)$$H_*(\RP^n; \Zz)$
• $H_*(\RP^n; \Zz_\omega)$$H_*(\RP^n; \Zz_\omega)$
• $H_*(S^{n-1} \tilde \times S^1; \Zz)$$H_*(S^{n-1} \tilde \times S^1; \Zz)$
• $H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$$H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$
• $H_*(\RP^n \sharp N; \Zz_\omega)$$H_*(\RP^n \sharp N; \Zz_\omega)$
• $H_*(*S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$$H_*(*S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$