Abstract
Let 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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Communicated by Claude LeBrun.
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Munn, M. Volume Growth and the Topology of Manifolds with Nonnegative Ricci Curvature. J Geom Anal 20, 723–750 (2010). https://doi.org/10.1007/s12220-010-9125-4
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DOI: https://doi.org/10.1007/s12220-010-9125-4