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 $M_\infty$.