Betti number: Difference between revisions

Latest revision as of 00:49, 28 July 2011

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

Definition

Given a topological space $X$ , the $n^{th}$ Betti number of $X$ , denoted $b_{n}(X)$ , is a nonnegative integer defined in any of the following equivalent ways. Note that if any of these definitions gives a finite number, so do all the others, and the values of the numbers are equal.:

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 (i.e., the quotient by the torsion subgroup). This makes sense if the torsion-free part is a finitely generated abelian group.
It is the dimension of the $n^{th}$ singular homology group $H_{n}(X;\mathbb {Q} )$ as a vector space over $\mathbb {Q}$ . This makes sense if the vector space is finite-dimensional.
It is the free rank of the $n^{th}$ singular cohomology group $H^{n}(X:\mathbb {Z} )$ , where free rank refers to the rank of the torsion-free part (i.e., the quotient by the torsion subgroup) as a free abelian group. This makes sense if the torsion-free part is a finitely generated abelian group.
It is the dimension of the $n^{th}$ singular cohomology group $H^{n}(X;\mathbb {Q} )$ as a vector space over $\mathbb {Q}$ . This makes sense if the vector space is finite-dimensional.

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 $-1$ .
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.

@@ Line 1: / Line 1: @@
-{{homotopy-invariant number}}
+{{homology-dependent invariant}}
 ==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 in any of the following equivalent ways. Note that if any of these definitions gives a finite number, so do all the others, and the values of the numbers are equal.:
+# It is the free rank of the <math>n^{th}</math> [[defining ingredient::singular homology]] group <math>H_n(X;\mathbb{Z})</math>, where ''free rank'' refers to the rank of the torsion-free part (i.e., the quotient by the torsion subgroup). This makes sense if the torsion-free part is a finitely generated abelian group.
+# 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>. This makes sense if the vector space is finite-dimensional.
+# It is the free rank of the <math>n^{th}</math> [[defining ingredient::singular cohomology]] group <math>H^n(X:\mathbb{Z})</math>, where '''free rank''' refers to the rank of the torsion-free part (i.e., the quotient by the torsion subgroup) as a free abelian group. This makes sense if the torsion-free part is a finitely generated abelian group.
+# It is the dimension of the <math>n^{th}</math> [[singular cohomology]] group <math>H^n(X;\mathbb{Q})</math> as a vector space over <math>\mathbb{Q}</math>. This makes sense if the vector space is finite-dimensional.
+==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]].