Poincare duality theorem: Difference between revisions
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<math>H^i(M;R) \to H_{n-i}(M;R)</math> | <math>H^i(M;R) \to H_{n-i}(M;R)</math> | ||
Poincare duality theorem states that this map is an isomorphism. | The '''Poincare duality theorem states''' that this map is an isomorphism. | ||
==Related | ==Related facts== | ||
===Similar facts=== | |||
* [[Alexander duality theorem]] | * [[Alexander duality theorem]] | ||
* [[Lefschetz duality theorem]] | * [[Lefschetz duality theorem]] | ||
===Applications=== | |||
* [[Euler characteristic of odd-dimensional compact connected orientable manifold is zero]] | |||
Latest revision as of 23:45, 21 July 2011
This article is about a duality theorem
Statement
Let be a compact connected orientable manifold. Choose to be fundamental class in . Then the cap product with defines a map:
The Poincare duality theorem states that this map is an isomorphism.