Remarks on Finite Subset Spaces

TL;DR

This paper investigates the topological properties of finite subset spaces of simplicial complexes, focusing on connectivity, homology, and the contractibility of certain subspaces, with new insights into their algebraic topology.

Contribution

It refines existing results on the topology of finite subset spaces, especially regarding their connectivity, homology groups, and the contractibility of singleton inclusions.

Findings

The three-fold subset space's inclusion of singletons is weakly contractible.

Finite subset spaces have specific connectivity and homology properties.

The subspace of singletons is generally non-contractible unless X is a co-H group.

Abstract

This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex $X$ including their connectivity and their top homology groups. It also discusses the inclusion of the singletons into the three fold subset space and shows that this subspace is weakly contractible but generally non-contractible unless $X$ is a co-$H$ group. Some homological calculations are provided.