Random Simple-Homotopy Theory

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 < 7, 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)-vertex triangulations of Bing's houses with k rooms, which all can be deformed to a point using only six pure elementary expansions.