Elementary Methods for Persistent Homotopy Groups

Adams H., Batan M. A., PAMUK M., Varlı H.

Discrete and Computational Geometry, 2025 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2025
  • Doi Number: 10.1007/s00454-025-00781-y
  • Journal Name: Discrete and Computational Geometry
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH, Civil Engineering Abstracts
  • Keywords: Excision theorem, Hurewicz theorem, Interleaving, Persistent fundamental group, Van Kampen theorem
  • Middle East Technical University Affiliated: Yes

Abstract

We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems. We prove a persistent excision theorem, derive from it a persistent Freudenthal suspension theorem, and obtain a persistent Hurewicz theorem relating the first nonzero persistent homotopy group of a space to its persistent homology. As an application, we compute sublevelset persistent homotopy groups of alkane energy landscapes and show these invariants capture nontrivial loops and higher-dimensional features that complement the information given by persistent homology.