Intrinsic flat convergence of covering spaces

Abstract

We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, \(M_j\), which converge to a nonzero integral current space, \(M_\infty \), in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact. we provide a sequence of simply connected \(M_j\) diffeomorphic to \({\mathbb {S}}^4\) that converge in the intrinsic flat sense to a torus \({\mathbb {S}}^1\times {\mathbb {S}}^3\). Nevertheless, we prove that if the \(\delta \)-covers, \({\tilde{M}}_j^\delta \), have finite order N, then a subsequence of the \({\tilde{M}}_j^\delta \) converge in the intrinsic flat sense to a metric space, \(M^\delta _\infty \), which is the disjoint union of covering spaces of \(M_\infty \).