Random simple-homotopy theory

Abstract

Abstract We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions . For triangulated d -manifolds with $$d\le 6$$ d ≤ 6 , we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, $$(14k+1)$$ ( 14 k + 1 ) -vertex triangulations of a new series of Bing’s houses with k rooms, $$k\ge 3$$ k ≥ 3 , which all can be deformed to a point using only six pure elementary expansions.