Gromov-Hausdorff Limits of Aspherical Manifolds

Abstract

Let $X$ be a compact Gromov-Hausdorff limit space of a collapsing sequence of compact $n$-manifolds, $M_i$, of Ricci curvature ${\text{Ric}}_{M_i}\ge -(n-1)$ and all points in $M_i$ are $(\delta ,\rho )$-local rewinding Reifenberg points, or sectional curvature ${\text{sec}}_{M_i}\ge -1$, respectively. We conjecture that if $M_i$ is an aspherical manifold of fundamental group satisfying a certain condition (e.g., a nilpotent group), then $X$ is a differentiable, or topological aspherical manifold, respectively. A main result in this paper asserts that if $M_i$ a diffeomorphic or homeomorphic to a nilmanifold, then $X$ is diffeomorphic or homeomorphic to a nilmanifold, respectively.