A counterexample to Wegner's conjecture on good covers

TL;DR

This paper disproves Wegner's 1975 conjecture by providing a counterexample, showing that the nerve of a finite good cover in R^d need not be d-collapsible, challenging previous assumptions in topological combinatorics.

Contribution

The authors construct a specific counterexample demonstrating that the nerve of a good cover in R^d can fail to be d-collapsible, refuting Wegner's long-standing conjecture.

Findings

Counterexample shows the nerve is not always d-collapsible

Disproves Wegner's conjecture from 1975

Impacts understanding of good covers in topology

Abstract

In 1975 Wegner conjectured that the nerve of every finite good cover in R^d is d-collapsible. We disprove this conjecture. A good cover is a collection of open sets in R^d such that the intersection of every subcollection is either empty or homeomorphic to an open d-ball. A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 which is contained in a unique maximal face.