Semi-local simple connectedness of non-collapsing Ricci limit spaces

TL;DR

The paper proves that non-collapsing Ricci limit spaces are semi-locally simply connected by showing small loops can be contracted within slightly larger neighborhoods.

Contribution

It establishes semi-local simple connectedness for non-collapsing Ricci limit spaces, a property previously unconfirmed in this context.

Findings

Every loop in a small ball is contractible in a slightly larger ball.

Non-collapsing Ricci limit spaces are semi-locally simply connected.

The result applies to all points in such spaces.

Abstract

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $\epsilon>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+\epsilon)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.