Describing the universal cover of a noncompact limit
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.
scientific article; zbMATH DE number 5826895