Contractible spaces and coalescent homotopies

TL;DR

This paper investigates the conditions under which contractible spaces admit coalescent contractions, providing criteria that distinguish spaces like the Dunce Hat and Bing's house where such contractions do not exist.

Contribution

It introduces the star-disc property as a criterion to determine the absence of coalescent contractions in contractible finite simplicial complexes.

Findings

Spaces like the Dunce Hat lack coalescent contractions.

The star-disc property characterizes complexes without coalescent contractions.

Collapse of simplicial complexes encodes coalescent contractions.

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.