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
:
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for the projective plane :
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for the torus :
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For an -dimensional complex
the sum
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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=32659
Betti number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Betti_number&oldid=32659
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article