Poincaré duality (Ex)
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# $H_*(\RP^n \sharp N; \Zz_\omega)$ | # $H_*(\RP^n \sharp N; \Zz_\omega)$ | ||
# $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ | # $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ | ||
+ | # $H_*(S^1 \times N; \Z_\omega)$ | ||
+ | # $H_*(\RP^n \times N; \Z_\omega)$ | ||
</wikitex> | </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 == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 17:11, 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
- $H^*(\RP^n; \Zz_\omega)$
- $H^*(\RP^n; \Zz)$
- $H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)$
- $H^*(S^{n-1} \tilde \times S^1; \Zz)$
- $H^*(\RP^n \sharp N; \Zz)$
- $H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)$
- $H^*(S^1 \times N; \Z)$
- $H^*(\RP^n \times N; \Z)$