Difference between revisions of "Betti number"
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The rank of the $r$-dimensional [[Betti group|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 $S^n$: | The rank of the $r$-dimensional [[Betti group|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 $S^n$: | ||
− | $$p^0=1,\quad p^1=\ | + | $$p^0=1,\quad p^1=\dotsb=p^{n-1}=0,\quad p^n=1;$$ |
for the projective plane $P^2(\mathbf R)$: | for the projective plane $P^2(\mathbf R)$: |
Latest revision as of 13:05, 14 February 2020
$r$-dimensional Betti number $p^r$ of a complex $K$
The rank of the $r$-dimensional Betti group with integral coefficients. For each $r$ the Betti number $p^r$ is a topological invariant of the polyhedron which realizes the complex $K$, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere $S^n$:
$$p^0=1,\quad p^1=\dotsb=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) |
Betti number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Betti_number&oldid=44611