On Elser's conjecture and the topology of $U$-nucleus complex

TL;DR

This paper proves a conjecture about the homology of the $U$-nucleus complex, building on prior work that connected Elser's conjecture to topological properties of this simplicial complex.

Contribution

It establishes the homology conjecture for the $U$-nucleus complex, advancing understanding of its topological structure and its relation to Elser's conjecture.

Findings

Proved the homology conjecture for $U$-nucleus complex.

Connected the homology properties to Elser's conjecture.

Enhanced understanding of the topological structure of $U$-nucleus complex.

Abstract

Dorpalen-Barry et al. proved Elser's conjecture about sign of Elser's number by interpreting them as certain sums of reduced Euler characteristics of an abstract simplicial complex known as $U$-nucleus complex. We prove a conjecture posed by them regarding the homology of $U$-nucleus complex.