A counterexample to Wegner's conjecture on good covers

Details A counterexample to Wegner's conjecture on good covers A counterexample to Wegner's conjecture on good covers

2010-08-11

arxiv.org/abs/1008.1895v1

Mathematics

Combinatorics

8 pages, 3 figures; recommended to print in color

Scientific paper

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.

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