Poincaré duality (Ex)
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− | Let $\Zz_\omega$ | + | Let $\Zz_\omega$ the local coefficient system of $\Zz$ 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): | Determine the following homology groups (in terms of $H_*(N; \Zz $ where appropriate): |
Revision as of 18:26, 9 February 2012
Let the local coefficient system of twisted the orientation character of a compact manifold , let denote the total space of the non-trivial linear sphere bundle over and let be a closed simply connected manifold.
Determine the following homology groups (in terms of where appropriate):
Determine the following cohomology groups and verify Poincaré duality using the homology computations above:
Here are some helpful references for the definitions involved: [Davis&Kirk2001, Ch 5] [Wall1967a, Chapter 1]