Gromov's Betty Number Theorem

Gromov's Betty number theorem gives an upper bound on the sum of the Betti numbers of a compact Riemannian manifold through the lower bound of its sectional curvatures, dimension and diameter.

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In particular, the sum of Betti numbers of a compact Riemannian manifold of dimension $n$ {\ displaystyle n} with non-negative sectional curvature is bounded by a constant $C$ $($ $n$ $)$ {\ displaystyle C (n)} .
- Supposedly $C(n)=2^{n}$ ${\ displaystyle C (n) = 2 ^ {n}}$ i.e. flat $n$ ${\ displaystyle n}$ -dimensional torus has the maximum sum of Betti numbers among all $n$ ${\ displaystyle n}$ -dimensional manifolds of non-negative sectional curvature.
- Explicit estimates are known, for example $C(n)=10^{3n^{4}+9n^{3}+6n^{2}}$ ${\ displaystyle C (n) = 10 ^ {3n ^ {4} + 9n ^ {3} + 6n ^ {2}}}$ .
The theorem gives an estimate for the Euler characteristic $n$ {\ displaystyle n} -dimensional manifold of non-negative sectional curvature.
- Presumably all such varieties have a non-negative Eulerian characteristic.

Literature

Gromov, Michael. Curvature, diameter and Betti numbers. (English) // Comment. Math. Helv. - 1981. - Vol. 56 , no. 2 . - P. 179–195 .

Gromov's Betty Number Theorem

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Literature

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