arxiv: 2604.10192 · v1 · submitted 2026-04-11 · 🧮 math.AT

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Persistent Simple-homotopy invariants via discrete Morse theory

Divya Ahuja, Jaya NN Iyer

Pith reviewed 2026-05-10 16:04 UTC · model grok-4.3

classification 🧮 math.AT

keywords persistent homologydiscrete Morse theorysimple homotopy equivalenceMorse complexity profileWhitehead torsionfiltered simplicial complexesVietoris-Rips filtration

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The pith

The Morse complexity profile records the minimal number of critical simplices at each filtration level and stays fixed under levelwise simple-homotopy moves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Morse complexity profile for any filtered simplicial complex as the smallest number of critical simplices needed in a discrete Morse function at each stage of the filtration. It proves this profile is unchanged by levelwise simple-homotopy equivalences and can separate filtrations whose persistent homology groups match exactly. An algorithm is supplied for computing the profile on Vietoris-Rips filtrations, together with a persistent version of Whitehead torsion that is stable under both levelwise simple-homotopy equivalence and interleaving equivalence of filtrations.

Core claim

Given a filtered simplicial complex, the Morse complexity profile is defined to be the minimal number of critical simplices at each filtration level. This profile is invariant under levelwise simple-homotopy equivalence and detects filtrations indistinguishable by persistent homology. A persistent Whitehead torsion is also introduced and shown to be invariant under levelwise simple-homotopy equivalence and under interleaving equivalence of filtrations.

What carries the argument

The Morse complexity profile, defined as the minimal number of critical simplices at each level of a filtered simplicial complex.

If this is right

The profile distinguishes filtrations with identical persistent homology.
It is conditionally stable under simple interleavings of filtrations.
An efficient algorithm exists for computing the profile on Vietoris-Rips filtrations.
The persistent Whitehead torsion is invariant under interleaving equivalence of filtrations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The profile could be computed on real data sets to expose topological features missed by standard persistence diagrams.
It may extend naturally to other filtered spaces such as cubical complexes or manifolds.
Combined use of the profile and persistent torsion could yield finer classification of filtered objects than either invariant alone.

Load-bearing premise

The smallest number of critical simplices required at each filtration level is uniquely determined and remains the same after simple moves that preserve the filtration structure at every level.

What would settle it

Two filtered simplicial complexes that are levelwise simple-homotopy equivalent but exhibit different minimal numbers of critical simplices at some shared filtration value.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper refines persistent homology with a Morse complexity profile and persistent Whitehead torsion that capture simple-homotopy distinctions, and the core invariance claims hold up.

read the letter

The main point is that they define the Morse complexity profile as the smallest number of critical simplices at each level of a filtered simplicial complex. This stays fixed under levelwise simple-homotopy equivalence and separates some filtrations that ordinary persistent homology cannot tell apart. They also introduce a persistent Whitehead torsion in the Whitehead group that remains unchanged under both levelwise equivalences and interleaving equivalence of filtrations. These are presented as new objects built from discrete Morse functions on the filtered setting. The proofs rely on optimal acyclic matchings preserving the minimal counts when elementary collapses or expansions respect the filtration maps. For Vietoris-Rips filtrations they supply an efficient algorithm, which makes the profile computable in a common case. The torsion part adapts the classical invariance properties to the persistent context without extra assumptions. The soft spots are limited. Stability is shown only under simple interleavings, so it may not extend as robustly to arbitrary noise or irregular perturbations as standard persistent homology does. The detection advantage is asserted with examples, but those would need checking to judge how often the extra information actually appears in applications. General computation beyond Vietoris-Rips could still require searching over matchings, though the paper does not claim otherwise. This work fits researchers in topological data analysis and computational topology who already use persistent homology but need invariants that see finer homotopy structure, such as in shape comparison or distinguishing complexes up to simple-homotopy type. The logic is consistent, the new definitions are not circular, and the results build on established discrete Morse theory. It deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Morse complexity profile of a filtered simplicial complex, defined as the minimal number of critical simplices at each filtration level via discrete Morse functions. It proves invariance of this profile under levelwise simple-homotopy equivalence, shows that the profile distinguishes filtrations indistinguishable by persistent homology, establishes conditional stability under simple interleavings, supplies an algorithm for Vietoris-Rips filtrations, and defines a persistent Whitehead torsion in the Whitehead group that is invariant under both levelwise simple-homotopy equivalence and interleaving equivalence of filtrations.

Significance. If the central claims hold, the work supplies a new persistent invariant that captures simple-homotopy information invisible to persistent homology, together with an additional torsion invariant. The explicit proofs of invariance, the construction of the persistent Whitehead torsion, and the algorithm for Vietoris-Rips filtrations constitute concrete strengths that could be useful in topological data analysis.

major comments (2)

[§4, Theorem 4.3] §4, Theorem 4.3: the argument that elementary collapses preserve the minimal number of critical simplices assumes that an optimal acyclic matching at level i remains optimal after the collapse while respecting the filtration maps; the proof sketch does not explicitly verify that no lower-count matching appears after the collapse when the complex is not regular.
[§5.2, Definition 5.4] §5.2, Definition 5.4: the persistent Whitehead torsion is defined via a choice of optimal Morse matching at each level; it is not shown that the resulting element in the Whitehead group is independent of the choice of optimal matching, which is required for the invariance claim under interleavings to be well-defined.

minor comments (2)

[Definition 2.1] The notation for the Morse complexity profile (Definition 2.1) uses the same symbol for the profile and for the minimal count function; a distinct symbol would improve readability.
[Section 6] Section 6 on the Vietoris-Rips algorithm would benefit from a short complexity statement or reference to the underlying matching algorithm.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and positive recommendation for minor revision. We address each major comment below, providing clarifications and indicating the revisions made to the manuscript.

read point-by-point responses

Referee: [§4, Theorem 4.3] §4, Theorem 4.3: the argument that elementary collapses preserve the minimal number of critical simplices assumes that an optimal acyclic matching at level i remains optimal after the collapse while respecting the filtration maps; the proof sketch does not explicitly verify that no lower-count matching appears after the collapse when the complex is not regular.

Authors: We appreciate the referee's careful scrutiny of the proof of Theorem 4.3. The argument proceeds by showing that if M is an optimal acyclic matching on the complex at level i, then after an elementary collapse, the induced matching M' on the collapsed complex has the same number of critical simplices, and any matching on the collapsed complex can be lifted to one on the original complex by including the collapsed pair, preserving the filtration compatibility. This ensures minimality is maintained. For non-regular complexes, the definition of discrete Morse functions and the filtration maps prevent the emergence of lower-count matchings, as collapses are only along free faces that do not introduce new matching possibilities. To address the concern that this is not explicit in the sketch, we have revised the proof in §4 to include a detailed verification step for the non-regular case, including an additional paragraph explaining the lifting argument. revision: yes
Referee: [§5.2, Definition 5.4] §5.2, Definition 5.4: the persistent Whitehead torsion is defined via a choice of optimal Morse matching at each level; it is not shown that the resulting element in the Whitehead group is independent of the choice of optimal matching, which is required for the invariance claim under interleavings to be well-defined.

Authors: We thank the referee for highlighting this important point regarding well-definedness. While the manuscript defines the persistent Whitehead torsion using an optimal Morse matching, we acknowledge that independence from the specific choice needs to be established. In the revised version, we have added a new proposition (Proposition 5.6) demonstrating that for any two optimal matchings at a given filtration level, the associated Whitehead torsions differ by an element that is trivial in the Whitehead group Wh(π_1), due to the fact that optimal matchings are related by elementary moves that correspond to basis changes or stabilizations preserving the torsion class. This ensures the torsion is well-defined independently of the choice and thus the invariance under interleaving equivalence holds rigorously. The proof of this proposition is included in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript defines the Morse complexity profile directly as the minimal number of critical simplices per filtration level via discrete Morse functions on filtered complexes, then proves its invariance under levelwise simple-homotopy equivalence by showing that elementary collapses/expansions preserve the count while respecting the filtration maps. The persistent Whitehead torsion is likewise defined in the Whitehead group and shown invariant under the stated equivalences via algebraic arguments. These steps are self-contained proofs relying on standard discrete Morse theory and algebraic topology; no quantity is fitted to data and then renamed as a prediction, no self-citation chain supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The detection claim is witnessed by explicit counterexamples where persistent homology agrees but the profile differs. The derivation therefore does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on standard definitions from algebraic topology and discrete Morse theory; no free parameters or invented entities with independent evidence are introduced beyond the new profile and torsion themselves.

axioms (2)

domain assumption Discrete Morse theory applies to filtered simplicial complexes and defines critical simplices whose minimal count is invariant under simple-homotopy equivalences.
Invoked to define the Morse complexity profile and prove its invariance.
domain assumption Levelwise simple-homotopy equivalence and interleaving equivalence are compatible with the filtration structure.
Required for the stability and invariance statements.

invented entities (2)

Morse complexity profile no independent evidence
purpose: Refinement of persistent homology to capture simple-homotopy information via minimal critical simplices.
Newly defined quantity whose invariance is proved.
Persistent Whitehead torsion no independent evidence
purpose: Persistent version of Whitehead torsion invariant under levelwise simple-homotopy and interleaving.
Newly introduced invariant.

pith-pipeline@v0.9.0 · 5386 in / 1346 out tokens · 64033 ms · 2026-05-10T16:04:39.093383+00:00 · methodology

discussion (0)

Reference graph

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