Contractible spaces and coalescent homotopies

Abstract

This paper deals with the existence, or absence, of coalescent contractions of contractible spaces. These are the contractions such that when the tracks of any two points meet, at time t0, they remain together thereafter. If a finite simplicial complex K is collapsible, then any collapse of K encodes coalescent contractions of K. Examples of contractible spaces where no coalescent contractions exist are the Dunce Hat and Bing's house. We establish a criteria for contractible finite simplicial complexes that ensures there are no coalescent contractions: the star-disc property. Keywords: contractible spaces, coalescent homotopies, dunce hat, Bing's house.