Poincaré duality (Ex)

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	<wikitex>;		<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.	+	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):
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	Here are some helpful references for the definitions involved: {{citeD|Davis&Kirk2001|Ch 5}} {{citeD|Wall1967a|Chapter 1}}		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]]

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Latest revision as of 14:55, 1 April 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): # $H_*(\RP^n; \Zz)$ # $H_*(\RP^n; \Zz_\omega)$ # $H_*(S^{n-1} \tilde \times S^1; \Zz)$ # $H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)$ # $H_*(\RP^n \sharp N; \Zz_\omega)$ # $H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)$ # $H_*(S^1 \times N; \Zz_\omega)$ # $H_*(\RP^n \times N; \Zz_\omega)$ 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> == References == {{#RefList:}} [[Category:Exercises]]\Zz_\omega$ the local coefficient system of $\Zz$ twisted the orientation character $\omega \colon \pi_1(M) \to \Zz/2$ of a compact manifold

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$M$, 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]