# Betti number

From Encyclopedia of Mathematics

*-dimensional Betti number of a complex *

The rank of the -dimensional Betti group with integral coefficients. For each the Betti number is a topological invariant of the polyhedron which realizes the complex , and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere :

for the projective plane :

for the torus :

For an -dimensional complex the sum

is equal to its Euler characteristic. Betti numbers were introduced by E. Betti [1].

#### References

[1] | E. Betti, Ann. Mat. Pura Appl. , 4 (1871) pp. 140–158 |

#### Comments

#### References

[a1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |

**How to Cite This Entry:**

Betti number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Betti_number&oldid=16078

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article