Volume growth and the topology of pointed Gromov-Hausdorff limits

In this paper 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 {(Min,pi)}i=1∞ with nonnegative Ricci curvature. Cheeger and Colding (1997) [7] showed that given such a sequence of Riemannian manifolds it is possible to define a measure ν 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 its volume growth. Namely, given constants α(k,n) which were computed in Munn (2010) [16] and based on earlier work of G. Perelman, we show that if limr→∞ν(Bp(r))ωnrn>α(k,n), then the kth homotopy group of (Y,p) is trivial. The constants α(k,n) are explicit and depend only on n, the dimension of the manifolds {(Min,pi)}, and k, the dimension of the homotopy in (Y,p).