Random simple-homotopy theory

• Published in: Journal of applied and computational topology Vol. 8; no. 3; pp. 531 - 555
• Main Authors: Benedetti, Bruno, Lai, Crystal, Lofano, Davide, Lutz, Frank H.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2024
• Subjects: Algebraic Topology; Computational Science and Engineering; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; 57Q35; 57M05; Bistellar flip; 57Q15; Simple-homotopy theory; Simplicial collapse and anticollapse; 57Q10; 57-04; Pure elementary expansion
• ISSN: 2367-1726; 2367-1734
• DOI: 10.1007/s41468-023-00139-4

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 ≤ 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, ( 14 k + 1 ) -vertex triangulations of a new series of Bing’s houses with k rooms, k ≥ 3 , which all can be deformed to a point using only six pure elementary expansions.