Persistent Simple-homotopy invariants via discrete Morse theory

Abstract: We introduce a refinement of persistent homology that detects simple-homotopy-theoretic phenomena invisible to homology. Given a filtered simplicial complex, we define the Morse complexity profile as the minimal number of critical simplices at each filtration level. We prove that this profile is invariant under levelwise simple-homotopy equivalence and detects filtrations indistinguishable by persistent homology. We establish conditional stability under simple interleavings and provide an efficient algorithm for Vietoris-Rips filtrations. We also introduce a persistent version of Whitehead torsion and show that it is invariant under both levelwise simple-homotopy equivalence and interleaving equivalence of filtrations.

• The paper introduces the Morse complexity profile, a novel invariant computed through discrete Morse theory that detects combinatorial obstructions overlooked by classical persistent homology.
• The method employs a greedy acyclic matching algorithm to approximate minimal Morse numbers efficiently, despite NP-hardness in exact computation.
• It also develops a persistent Whitehead torsion invariant that distinguishes non-collapsible structures, enhancing the topological toolkit beyond abelian invariants.

Persistent Simple-Homotopy Invariants via Discrete Morse Theory

Introduction and Motivation

This work presents a novel refinement of persistent homology—the Morse complexity profile—rooted in discrete Morse theory, designed to detect simple-homotopy-theoretic phenomena that are invisible to classical homological invariants. Persistent homology, while foundational in TDA and robust to perturbations, only encodes information accessible through abelian invariants, missing finer combinatorial obstructions such as non-collapsibility or distinctions beyond classical homotopy equivalence. This limitation is circumvented by importing tools from discrete Morse theory and Whitehead's theory of simple-homotopy, yielding new computable invariants with significant theoretical and computational implications.

Morse Complexity Profile: Definition and Invariance Properties

The Morse complexity profile $\mathcal{M}(K_\bullet)$ associates to a filtered simplicial complex $K_\bullet$ the minimal Morse number at each filtration index, where the minimal Morse number $M(K)$ is the minimal total number of critical simplices (across all discrete Morse functions). This encapsulates information about collapsibility at each scale and detects combinatorial changes in structure which are not reflected in homology.

A central theorem establishes that $M(K)$ is a simple-homotopy invariant: if two finite simplicial complexes are simple-homotopy equivalent, then their minimal Morse numbers and the dimension-wise counts of critical simplices coincide. This property extends to filtrations—levelwise simple-homotopy equivalence ensures invariance of the full Morse complexity profile.

Morse Spikes and Distinction Beyond Persistent Homology

Filtration levels where the minimal Morse number increases despite unchanged homology are termed Morse spikes. These signal transitions where collapsibility temporarily fails, corresponding to non-trivial combinatorial obstructions without any manifestation in persistent homology or Betti number evolution.

The authors prove the existence of filtered complexes $\{K_0\subset K_1\subset K_2\}$ which are indistinguishable by persistent homology (each $K_i$ is contractible and all induced maps are isomorphisms on homology), but exhibit nontrivial Morse spikes, explicitly constructing such a sequence using the Dunce hat—the canonical example of a contractible but non-collapsible complex. This decisively demonstrates that persistent homology fails to detect certain fundamental topological phenomena.

Computational Realizability and Algorithmic Aspects

A greedy acyclic matching algorithm is developed to compute the number of critical simplices at each filtration step with $O(N)$ total time overhead over the baseline cost of constructing the filtration, where $N$ is the total number of simplices across the filtration. While computing the minimal Morse number is NP-hard, this approach yields practical, scalable approximations of the Morse complexity profile, readily applicable to filtrations such as Vietoris-Rips complexes arising in TDA pipelines.

Persistent Whitehead Torsion: A Finer Invariant

Pushing beyond Morse-theoretic invariants, the authors propose a persistent Whitehead torsion invariant. They generalize Whitehead torsion to the persistent setting by associating to windows of filtration indices with levelwise homotopy equivalences a $1$-cocycle in the Whitehead group of the common fundamental group. They show this assignment is well-defined and invariant under both levelwise simple-homotopy equivalence and suitable interleaving (even up to homotopy), yielding a persistent analogue of the classical obstruction to simple-collapsibility.

The development of the persistent torsion cocycle allows rigorous obstruction-theoretic tracking of the failure of inclusions between filtration steps to be simple homotopy equivalences. When homology alone stabilizes, persistent Whitehead torsion can still record persistent, essential differences in combinatorial structure.

Stability and Hierarchy of Invariants

The authors prove conditional stability of the Morse complexity profile for simple interleavings—interleavings where the connecting maps are simple homotopy equivalences—with Morse spikes persisting up to a bounded filtration index shift. Persistent Whitehead torsion is similarly shown to be invariant under $\epsilon$-interleaving by homotopy equivalences, modulo coherent homotopies.

The work thus positions TDA invariants in a strict hierarchy:

$K_\bullet$0

with each inclusion strict, and each layer capable of detecting strictly more refined phenomena than the one below.

Implications and Future Directions

The persistent Morse complexity profile and persistent Whitehead torsion provide computable, stable, and theoretically grounded tools for probing features of filtered complexes inaccessible to homological methods. They offer a path for TDA, applied topology, and data science practitioners to move beyond abelian invariants and engage with the full spectrum of combinatorial and homotopical complexity present in topological data.

From a theoretical standpoint, this framework invites extensions to more general cell complexes, interactions with algebraic $K_\bullet$1-theory, and possibly categorical TDA. From an applied perspective, the proposed algorithms are tractable for datasets of practical size, especially for those producing Vietoris-Rips or similar filtrations. An immediate direction for future research is the integration of these invariants into TDA pipelines and investigating their sensitivity to topological noise and model structure in real-world data.

Conclusion

This work develops and comprehensively studies new simple-homotopy invariants of filtered complexes—the Morse complexity profile and persistent Whitehead torsion—that strictly refine persistent homology. The methodology combines computability, theoretical robustness, and practical relevance for modern computational and applied topology. The introduction of these invariants marks an important expansion of the topological toolkit available for the study of complex data and filtered spaces, with future potential for both mathematical theory and data science applications.

Reference:

"Persistent Simple-homotopy invariants via discrete Morse theory" (2604.10192)