Elementary Methods for Persistent Homotopy Groups

TL;DR

This work develops elementary, computable methods for persistent homotopy, introducing $(k,l)$-persistent analogues of fundamental groups and higher homotopy, and proving persistent versions of the Van Kampen, excision, suspension, and Hurewicz theorems. It also defines an interleaving distance for persistent homotopy and shows stability results that relate the persistence of a whole space to the persistence of its parts. The theoretical framework is complemented by a substantive application to alkane energy landscapes, where sublevelset persistent homotopy groups reveal loops and higher-dimensional features that persist beyond what persistent homology captures. Together, these results extend homotopy-theoretic tools to persistent settings and demonstrate their value for analyzing complex energy landscapes and other filtered spaces.

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.

Figures (23)

• Figure 1: Commutative diagram of group homomorphisms.
• Figure 2: This diagram is commutative: $\pi_1(A\hookrightarrow X) \pi_1(A\cap B\hookrightarrow A) = \pi_1(B\hookrightarrow X) \pi_1(A\cap B\hookrightarrow B)$.
• Figure 3: Fundamental group homomorphisms between levels $k$ and $l$.
• Figure 4: This diagram is commutative: $j^A_{kl}\circ i^A_{kl} = j^B_{kl}\circ i^B_{kl}$.
• Figure 5: Pushout square of fundamental groupoids.

Theorems & Definitions (26)

• Definition 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5: Van Kampen theorem for the fundamental groupoid
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3