Poincare duality space: Difference between revisions
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{{homology-dependent topospace property}} | |||
==Definition== | ==Definition== | ||
Let <math>M</math> a [[connected space]] and <math>R</math> a commutative ring. We say that <math>M</math> is a '''Poincare duality space''' of formal dimension <math>n</math> with respect to <math>R</math> if the following hold: | Let <math>M</math> a [[connected space]] and <math>R</math> a commutative ring. We say that <math>M</math> is a '''Poincare duality space''' of formal dimension <math>n</math> with respect to <math>R</math> if the following hold: | ||
* The homology of <math>M</math> is finitely generated | * The homology of <math>M</math> with coefficients in <math>R</math> is finitely generated | ||
* <math>H_n(M;R)</math> is a free module of rank <math>1</math> over <math>R</math> | * <math>H_n(M;R)</math> is a free module of rank <math>1</math> over <math>R</math> | ||
* Pick a generator for <math>H_n(M;R)</math>. Then the [[cap product]] with this generator induces a map from <math>H^i(M)</math> to <math>H_{n-i}(M)</math>. This map is an isomorphism for all <math>i</math>. | * Pick a generator for <math>H_n(M;R)</math>. Then the [[cap product]] with this generator induces a map from <math>H^i(M)</math> to <math>H_{n-i}(M)</math>. This map is an isomorphism for all <math>i</math>. | ||
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In particular, <math>M</math> has nonvanishing homology groups only for <math>0 \le i \le n</math> | In particular, <math>M</math> has nonvanishing homology groups only for <math>0 \le i \le n</math> | ||
By default, we take <math>R = \mathbb{Z}</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 19:57, 11 May 2008
This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spaces
View all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces
Definition
Let a connected space and a commutative ring. We say that is a Poincare duality space of formal dimension with respect to if the following hold:
- The homology of with coefficients in is finitely generated
- is a free module of rank over
- Pick a generator for . Then the cap product with this generator induces a map from to . This map is an isomorphism for all .
In particular, has nonvanishing homology groups only for
By default, we take .
