Betti number: Difference between revisions

Revision as of 18:51, 2 April 2011

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

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

It is the free rank of the $n^{th}$ singular homology group $H_{n}(X;\mathbb {Z} )$ , where free rank refers to the rank of the torsion-free part. This makes sense if the $n^{th}$ 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 $n^{th}$ singular homology group $H_{n}(X;\mathbb {Q} )$ as a vector space over $\mathbb {Q}$ .

For a space with finitely generated homology, the ordinary generating function of the Betti numbers is a polynomial. This polynomial is termed the Poincare polynomial.
For a space with finitely generated homology, the signed sum of the Betti numbers is termed the Euler characteristic. This can also be viewed as the number obtained by evaluating the Poincare polynomial at $-1$ .
For space with homology of finite type, the ordinary generating function of the Betti numbers is a power series. This power series is termed the Poincare series.

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 # 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>.
+==Related notions==
+* For a [[space with finitely generated homology]], the ordinary generating function of the Betti numbers is a polynomial. This polynomial is termed the [[Poincare polynomial]].
+* For a [[space with finitely generated homology]], the signed sum of the Betti numbers is termed the [[Euler characteristic]]. This can also be viewed as the number obtained by evaluating the Poincare polynomial at <math>-1</math>.
+* For [[space with homology of finite type]], the ordinary generating function of the Betti numbers is a power series. This power series is termed the [[Poincare series]].