Betti number
From Encyclopedia of Mathematics
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-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
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