Betti number: Difference between revisions
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==Definition== | ==Definition== | ||
Given a topological space <math>X</math>, the <math>n^{th}</math> Betti number of <math>X</math> is defined as the rank of the <math>n^{th}</math> [[singular homology]] group (rank here is as in the rank of a [[free Abelian group]]). Here, we take the singular homology theory over <math>\mathbb{Z}</math>. | Given a topological space <math>X</math>, the <math>n^{th}</math> Betti number of <math>X</math> is defined as the rank of the <math>n^{th}</math> [[singular homology]] group (rank here is as in the rank of a [[free Abelian group]]). Here, we take the singular homology theory over <math>\mathbb{Z}</math>. | ||
Revision as of 19:51, 27 October 2007
This article describes an invariant of topological spaces that depends only on its homology groups
Definition
Given a topological space , the Betti number of is defined as the rank of the singular homology group (rank here is as in the rank of a free Abelian group). Here, we take the singular homology theory over .
