First Betti number and collapse

TL;DR

This paper investigates the behavior of the first Betti number in sequences of Riemannian manifolds collapsing under a lower Ricci curvature bound, establishing a limit on how much it can decrease.

Contribution

It proves that the first Betti number cannot decrease by more than the dimension during such collapsing sequences.

Findings

First Betti number is bounded below by the dimension during collapse.

Collapse under Ricci curvature bounds constrains topological complexity.

Provides new insights into the topology of collapsing manifolds.

Abstract

We show that when a sequence of Riemannian manifolds collapses under a lower Ricci curvature bound, the first Betti number cannot drop more than the dimension.