Volume growth and the topology of Gromov-Hausdorff limits

Abstract

We examine topological properties of pointed metric measure spaces $(Y,p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds $\{(M_i^n,p_i){\}}_{i=1}^{\infty }$ with nonnegative Ricci curvature. Cheeger and Colding \cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $\nu$ on the limit space $(Y,p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y,p)$ and the volume growth of $\nu$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if ${\lim }_{r\to \infty }\frac{\nu (B_p(r))}{{\omega }_n{r}^n}>\alpha (k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $\alpha (k,n)$ are explicit and depend only on $n$, the dimension of the manifolds $\{(M_i^n,p_i)\}$, and $k$, the dimension of the homotopy in $(Y,p)$.

14 pages. To appear in Differential Geometry and Its Applications.

References in corpus (1)

• The Topology of Open Manifolds with Nonnegative Ricci Curvature