Random Simple-Homotopy Theory

TL;DR

The paper introduces RSHT, a novel algorithm combining collapses and expansions to analyze simple-homotopy types of complexes, with applications to contractibility and substructure detection in high-dimensional spaces.

Contribution

It presents RSHT, an innovative algorithm that unifies elementary collapses and expansions for studying homotopy types, extending to triangulated manifolds and complex structures.

Findings

RSHT reduces to bistellar flips for d<7 triangulated manifolds.

Successfully triangulated complex examples like the Abalone and Bing's houses.

Deformation to a point achieved with minimal elementary expansions.

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.