Elementary Methods for Persistent Homotopy Groups

TL;DR

This paper develops foundational and computational methods for persistent homotopy groups, extending classical theorems to the persistent setting and applying these to analyze complex energy landscapes.

Contribution

It introduces elementary computational techniques and persistent analogues of key theorems like Van Kampen, excision, suspension, and Hurewicz, advancing the analysis of persistent homotopy groups.

Findings

Persistent homotopy groups capture nontrivial loops and higher features.

New persistent theorems extend classical algebraic topology results.

Application to alkane energy landscapes demonstrates practical utility.

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 comple- ment the information given by persistent homology.