Poincaré duality (Ex)
From Manifold Atlas
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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 | + | 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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# $H_*(\RP^n \sharp N; \Zz_\omega)$ | # $H_*(\RP^n \sharp N; \Zz_\omega)$ | ||
# $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ | # $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ | ||
+ | # $H_*(S^1 \times N; \Zz_\omega)$ | ||
+ | # $H_*(\RP^n \times N; \Zz_\omega)$ | ||
+ | |||
+ | Determine the following cohomology groups and verify Poincaré duality using the homology computations above: | ||
+ | # $H^*(\RP^n; \Zz_\omega)$ | ||
+ | # $H^*(\RP^n; \Zz)$ | ||
+ | # $H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)$ | ||
+ | # $H^*(S^{n-1} \tilde \times S^1; \Zz)$ | ||
+ | # $H^*(\RP^n \sharp N; \Zz)$ | ||
+ | # $H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)$ | ||
+ | # $H^*(S^1 \times N; \Zz)$ | ||
+ | # $H^*(\RP^n \times N; \Zz)$ | ||
+ | |||
+ | Here are some helpful references for the definitions involved: {{citeD|Davis&Kirk2001|Ch 5}} {{citeD|Wall1967a|Chapter 1}} | ||
</wikitex> | </wikitex> | ||
− | == References == | + | <!-- == References == |
− | {{#RefList:}} | + | {{#RefList:}} --> |
[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises without solution]] |
Latest revision as of 14:55, 1 April 2012
Tex syntax error, 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]