Regularity of Resolutions and Limits of Manifolds with a Uniform Contractibility Function

TL;DR

This paper proves a 1991 conjecture about the structure of Gromov--Hausdorff limits of manifolds with uniform contractibility, providing new characterizations and applications in topology and geometric analysis.

Contribution

It offers a self-contained proof of Moore's conjecture, characterizes limits of certain manifolds, and introduces obstructions for approximating homology manifolds by PL-manifolds.

Findings

Characterization of finite-dimensional Gromov--Hausdorff limits of manifolds

Obstructions to approximating homology manifolds by PL-manifolds

Applications to homology manifolds, Alexandrov spaces, and Wasserstein spaces

Abstract

In this paper, we give a short and self-contained proof to a 1991 conjecture by Moore concerning the structure of certain finite-dimensional Gromov--Hausdorff limits, in the ANR setting. As a consequence, one easily characterizes finite dimensional limits of PL-able or Riemannian $n$-manifolds with a uniform contractibility function. For example, one can define for any compact connected metric space that is a resolvable ANR homology manifold of covering dimension at least 5, an obstruction, which vanishes if and only if the homology manifold can be approximated in the Gromov--Hausdorff sense by PL-manifolds of the same dimension and with a uniform contractibility function. Further, it provides short proofs to certain well known results by reducing them to problems in Bing topology. We also give another proof using more classical arguments that yield more structural information. We give several applications to the theory of homology manifolds, Alexandrov spaces, Wasserstein spaces and a generalized form of the diffeomorphism stability conjecture.