Poincaré duality (Ex)
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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. | 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 | + | Determine the following homology groups (in terms of $H_*(N; \Zz $ where appropriate): |
# $H_*(\RP^n; \Zz)$ | # $H_*(\RP^n; \Zz)$ | ||
# $H_*(\RP^n; \Zz_\omega)$ | # $H_*(\RP^n; \Zz_\omega)$ | ||
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# $H_*(\RP^n \times N; \Zz_\omega)$ | # $H_*(\RP^n \times N; \Zz_\omega)$ | ||
| − | Determine the following cohomology groups and verify Poincaré duality using the homology computations above | + | Determine the following cohomology groups and verify Poincaré duality using the homology computations above: |
# $H^*(\RP^n; \Zz_\omega)$ | # $H^*(\RP^n; \Zz_\omega)$ | ||
# $H^*(\RP^n; \Zz)$ | # $H^*(\RP^n; \Zz)$ | ||
Revision as of 18:25, 9 February 2012
Let 
denote homology with local co-efficients in 
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]
