Intrinsic Flat Convergence of Covering Spaces

TL;DR

This paper investigates the behavior of covering spaces and spectra under intrinsic flat convergence of Riemannian manifolds, revealing that convergence of the manifolds does not necessarily imply convergence of their covering spaces, but under certain conditions, subsequences do converge.

Contribution

It demonstrates that covering spaces and spectra may not converge under intrinsic flat limits, but establishes conditions for subsequential convergence of $ ilde{M}_j^ ext{delta}$ covers.

Findings

Covering spectra need not converge under intrinsic flat convergence.

Subsequences of $ ilde{M}_j^ ext{delta}$ converge to disjoint unions of covering spaces.

Examples include convergence from spheres to a torus in the intrinsic flat sense.

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 $M_\infty$.