Volume Growth and the Topology of Manifolds with Nonnegative Ricci Curvature
2010; Springer Science+Business Media; Volume: 20; Issue: 3 Linguagem: Inglês
10.1007/s12220-010-9125-4
ISSN1559-002X
Autores Tópico(s)Homotopy and Cohomology in Algebraic Topology
ResumoLet M n be a complete, open Riemannian manifold with Ric≥0. In 1994, Grigori Perelman showed that there exists a constant δ n >0, depending only on the dimension of the manifold, such that if the volume growth satisfies $\alpha_{M}:=\lim_{r\rightarrow \infty}\frac{\operatorname{Vol}(B_{p}(r))}{\omega_{n}r^{n}}\geq 1-\delta_{n}$ , then M n is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, α(k,n), depending only on k and n, which guarantee the individual k-homotopy group of M n is trivial.
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