Betti number

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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.:

  1. 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.
  2. 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.
  3. 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.
  4. 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.

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