Describing the universal cover of a noncompact limit
type: Item
instance of: scholarly article
title: Describing the universal cover of a noncompact limit
zbMATH Open document ID: 1207.53048
author: Guofang Wei
author: J. Ennis
published in: Geometry \& Topology
publication date: 2010-12-15T00:00:00Z
review text: The notion of pseudo-nullhomotopic loops is introduced. A description of the universal cover of a noncompact limit space in terms of the covering spaces of balls of increasing size in the sequence is given by the following result: Theorem. Suppose \((X,x)= \text{GH\,lim}_{i\to\infty} (M_i,p_i)\), where \(M_i\) are \(n\)-dimensional Riemannian manifolds with a uniform lower bound on the Ricci curvature and GH lim denotes the Gromov-Hausdorff limit. Suppose further that for every loop \(j\) that lifts to a loop in \(\widetilde X\), there is \(R(j)\) such that \(j\) is pseudo-nullhomotopic in the ball \(B(x,R(j))\). Then, if \(\pi:\widetilde X\to X\) is the universal cover with \(\pi(\widetilde x)= x\), there are an increasing sequence \(R_i\to\infty\) and numbers \(\delta_i> 0\) such that \[ (\widetilde X,\widetilde x)= \underset{i\to\infty}{}{\text{GH\,lim}}(\widetilde B(p_i, R_i, R_{i+1})^{\delta_i},\widetilde p_i), \] where we pass to a subsequence of the manifolds \(M_i\) as well.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 5826895
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Ricci curvature
zbMATH Keywords: universal cover
zbMATH Keywords: Gromov-Hausdorff limit
zbMATH Keywords: compact length space
reviewed by: Mihail B. Banaru
MaRDI profile type: Publication
full work available at URL: gt.2010.14.2479
OpenAlex ID: W2045808945
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DOI: 10.2140/GT.2010.14.2479
description: scientific article; zbMATH DE number 5826895