Order complexes of coset posets of finite groups are not contractible

TL;DR

This paper proves that the order complex of the coset poset of any finite group is never contractible, answering a longstanding question and revealing topological properties of these algebraic structures.

Contribution

It establishes that the order complex of the coset poset of finite groups is never contractible, settling a question posed by K. S. Brown.

Findings

Order complex of coset poset is never $ ext{F}_2$-acyclic.

The result applies to all finite groups.

It confirms the non-contractibility of these complexes.

Abstract

We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never $\mathbb{F}_{2}$-acyclic and therefore never contractible. This settles a question of K. S. Brown.