On the universal cover and the fundamental group of an $RCD^*(K,N)$-space

TL;DR

This paper proves the existence of universal covers for $RCD^*(K,N)$-spaces, extending previous results on Ricci limit spaces, and derives new topological insights about their fundamental groups without additional assumptions.

Contribution

It establishes the existence of universal covers for $RCD^*(K,N)$-spaces and provides initial topological results on their fundamental groups without extra assumptions.

Findings

Universal cover exists for $RCD^*(K,N)$-spaces.

New structure results on the fundamental group.

First topological results without extra assumptions.

Abstract

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).