Poincaré duality II (Ex)
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Consider $S^1$ as a simplicial complex with three $0$-simplices and three $1$-simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map $$ -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)$$ thus verifying that $S^1$ has Poincaré duality. | Consider $S^1$ as a simplicial complex with three $0$-simplices and three $1$-simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map $$ -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)$$ thus verifying that $S^1$ has Poincaré duality. | ||
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[[Category:Exercises]] | [[Category:Exercises]] | ||
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Latest revision as of 14:56, 1 April 2012
For a simplicial complex , define the front -face of an -simplex as and the back -face as
The Alexander-Whitney diagonal approximation is given by
and the partial evaluation map is defined as
Recall, we define the cap product on the chain level by
and this descends to a well defined product on (co)homology.
Consider as a simplicial complex with three -simplices and three -simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map
thus verifying that has Poincaré duality.
$-simplices and three X, define the front -face of an -simplex as and the back -face as
The Alexander-Whitney diagonal approximation is given by
and the partial evaluation map is defined as
Recall, we define the cap product on the chain level by
and this descends to a well defined product on (co)homology.
Consider as a simplicial complex with three -simplices and three -simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map
thus verifying that has Poincaré duality.
$-simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map $$ -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)$$ thus verifying that $S^1$ has Poincaré duality.
== References ==
{{#RefList:}}
[[Category:Exercises]]X, define the front -face of an -simplex as and the back -face as
The Alexander-Whitney diagonal approximation is given by
and the partial evaluation map is defined as
Recall, we define the cap product on the chain level by
and this descends to a well defined product on (co)homology.
Consider as a simplicial complex with three -simplices and three -simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map
thus verifying that has Poincaré duality.