Betti number: Difference between revisions

Revision as of 18:49, 2 April 2011

This article describes an invariant of topological spaces that depends only on its homology groups

Given a topological space $X$ , the $n^{t h}$ Betti number of $X$ , denoted $b_{n} (X)$ , is a nonnegative integer defined as follows:

It is the free rank of the $n^{t h}$ singular homology group $H_{n} (X; Z)$ , where free rank refers to the rank of the torsion-free part. This makes sense if the $n^{t h}$ singular homology group is finitely generated, or more generally, if its quotient by its torsion subgroup is finitely generated.
It is the dimension of the $n^{t h}$ singular homology group $H_{n} (X; Q)$ as a vector space over $Q$ .

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 ==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>, denoted <math>b_n(X)</math>, is a nonnegative integer defined as follows:
+# It is the free rank of the <math>n^{th}</math> [[singular homology]] group <math>H_n(X;\mathbb{Z})</math>, where ''free rank'' refers to the rank of the torsion-free part. This makes sense if the <math>n^{th}</math> singular homology group is finitely generated, or more generally, if its quotient by its torsion subgroup is finitely generated.
+# It is the dimension of the <math>n^{th}</math> [[singular homology]] group <math>H_n(X;\mathbb{Q})</math> as a vector space over <math>\mathbb{Q}</math>.