On the local topology of non-collapsed Ricci bounded limit spaces

TL;DR

This paper proves that in non-collapsed Ricci limit spaces, the first Betti number of the regular parts is zero, providing insights into their local topology and raising new questions.

Contribution

It establishes the vanishing of the local first Betti number in regular loci of non-collapsed Ricci limit spaces, advancing understanding of their topological structure.

Findings

The local first Betti number of regular loci is zero.

Provides applications and discusses open questions in Ricci limit spaces.

Abstract

We show that for a pointed Gromov-Hausdorff limit of non-collapsed Riemannian manifolds with bounded Ricci curvature, the local $b_1$ of the regular loci vanishes. We also discuss applications and some open questions.