On the connectivity of finite subset spaces

TL;DR

This paper investigates the connectivity properties of finite subset spaces of a given cell complex, establishing a relation between the connectivity of the complex and its finite subset spaces.

Contribution

It proves that for an m-connected cell complex, the space of non-empty subsets of size at most k is (m + k - 2)-connected, extending understanding of subset space topology.

Findings

Finite subset spaces of an m-connected complex are (m + k - 2)-connected.

Connectivity increases predictably with subset size.

Results apply to a broad class of cell complexes.

Abstract

We show that for an m-connected cell complex X the space exp_k X of non-empty subsets of X of cardinality at most k is (m + k - 2)-connected