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, $M_{\infty }^{\delta }$, which is the disjoint union of covering spaces of $M_{\infty }$.

37 pages, 4 figures