Poincaré duality (Ex)

Revision as of 17:18, 9 February 2012

Let $<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 # $H_*(\RP^n; \Zz)$ # $H_*(\RP^n; \Zz_\omega)$ # $H_*(S^{n-1} \tilde \times S^1; \Zz)$ # $H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$ # $H_*(\RP^n \sharp N; \Zz_\omega)$ # $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ # $H_*(S^1 \times N; \Z_\omega)$ # $H_*(\RP^n \times N; \Z_\omega)$ </wikitex> Determine the following cohomology groups and verify Poincaré duality using the homology computations above # $H^*(\RP^n; \Zz_\omega)$ # $H^*(\RP^n; \Zz)$ # $H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)$ # $H^*(S^{n-1} \tilde \times S^1; \Zz)$ # $H^*(\RP^n \sharp N; \Zz)$ # $H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)$ # $H^*(S^1 \times N; \Z)$ # $H^*(\RP^n \times N; \Z)$ == References == {{#RefList:}} [[Category:Exercises]]\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

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

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

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

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

References