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Betti number

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$r$-dimensional Betti number $p^r$ of a complex $K$

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

$$p^0=1,\quad p^1=\ldots=p^{n-1}=0,\quad p^n=1;$$

for the projective plane $P^2(\mathbf R)$:

$$p^0=1,\quad p^1=p^2=0;$$

for the torus $T^2$:

$$p^0=p^2=1,\quad p^1=2.$$

For an $n$-dimensional complex $K^n$ the sum

$$\sum_{k=0}^n(-1)^kp^k$$

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://www.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