pith. sign in

arxiv: 2502.06520 · v3 · submitted 2025-02-10 · 🧮 math.CO · math.AT· math.RA

Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators

Anupam Mondal , Sajal Mukherjee , Pritam Chandra Pramanik This is my paper

Pith reviewed 2026-05-23 03:43 UTC · model grok-4.3

classification 🧮 math.CO math.ATmath.RA
keywords discrete Morse theorycritical pair cancellationboundary operatorsgradient vector fieldsimplicial homologycoboundary operatorschain complexes
0
0 comments X

The pith

An explicit formula expresses the modified boundary operators after cancelling a critical pair in discrete Morse theory using only the original operators.

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

The paper derives a direct algebraic formula that updates the boundary operators of a chain complex when a critical pair is cancelled in a discrete Morse function. This avoids the need to recount gradient trajectories after the cancellation. The same update can be achieved by performing elementary row operations on the original boundary matrices. A parallel result holds for the coboundary operators. If correct, this makes homology computations more efficient by reducing the reliance on trajectory enumeration.

Core claim

Cancelling an admissible critical pair perturbs the gradient trajectories, but the effect on the boundary operator can be captured exactly by an algebraic formula relating the new boundary operator to the old one. The formula is obtained combinatorially without reference to the new trajectories, and equivalently by a sequence of elementary row operations on the matrix of the original boundary operator.

What carries the argument

The explicit formula for the modified boundary operator in terms of the original boundary operator after cancelling a critical pair.

If this is right

  • The need to enumerate new gradient trajectories after cancellation is eliminated.
  • Homology groups of the simplified complex can be computed directly from the updated operators.
  • The same algebraic update applies to coboundary operators for cohomology.
  • Computations reduce to matrix operations on the original boundary data.

Where Pith is reading between the lines

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

  • Algorithms for discrete Morse theory could implement this update as a matrix manipulation step rather than a search over paths.
  • This relation might extend to other reductions or modifications of gradient vector fields beyond single pair cancellations.
  • The row operation view suggests connections to Gaussian elimination in chain complexes.

Load-bearing premise

The cancellation must be admissible for the discrete Morse function, and all changes to gradient trajectories must be accounted for exactly by the stated algebraic relation between old and new operators.

What would settle it

Take a simplicial complex with a known discrete Morse function having an admissible critical pair, compute the boundary operators before and after cancellation by direct trajectory enumeration, and check whether they satisfy the claimed explicit formula.

Figures

Figures reproduced from arXiv: 2502.06520 by Anupam Mondal, Pritam Chandra Pramanik, Sajal Mukherjee.

Figure 1
Figure 1. Figure 1: P is a V-trajectory (red) that starts from σ0, follows along P0 (black) to βs and then leaves P0 to reach τi . (3) Any W-trajectory from σj to σ0 first meets P0 at some (k − 1)-simplex αt , where t ∈ {1, . . . , r + 1} (see [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: P is a W-trajectory (red) from σj to σ0, which meets P0 (black) at αt , then follows along P0 to σ0. (4) Any V-trajectory from σj to τi , which involves some simplices from P0 has the following property. After starting from σj , it meets P0 at some (k − 1)-simplex αt , where t ∈ {1, . . . , r}, then follows along P0 to some k-simplex βs, with t ≤ s ≤ r, and then leaves P0 and ends at τi (see [PITH_FULL_IM… view at source ↗
Figure 3
Figure 3. Figure 3: P (red) is a V-trajefory from σj to τi that meets P0 (black) at αt , follows along P0 to βs and then leaves the trajectory P0 to reach τi . 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: P (red) is a W-trajectory from σj to τi that meets P0 (black) at αt , follows along P0 to βs and then leaves the trajectory P0 to reach τi . Lemma 3.3. For any σj ∈ CritW k (∆), X P:P is a W-traj. from σj to σ0 w(P) = −w(P0) · X P:P is a V-traj. from σj to τ0 w(P). Proof. Any V-trajectory P from σj to τ0 first meets P0 at some (k−1)-simplex, say αt , for some t ∈ {1, . . . , r + 1}, and then follows along … view at source ↗
Figure 5
Figure 5. Figure 5: P (blue) is a V-trajectory from σj to τ0, which meets P0 at αt . Then σjP αtP0σ0 (red) is the W-trajectory corresponding to P. Taking sum of the weights of all W-trajectories from σj to σ0, we get the desired result. Lemma 3.4. For any σj ∈ CritW k (∆) and τi ∈ CritW k−1 (∆), the sum of the weights of all W-trajectories from σj to τi , which include some simplices from the trajectory P0, is given by X P:P … view at source ↗
Figure 6
Figure 6. Figure 6: P1 (red) is a W-trajectory from σj to σ0 which meets P0 at αt . P2 (blue) is a V￾trajectory from σ0 to τi which leaves P0 at βs. For s ≥ t, φ((P1, P2)) = σjP1αtP0βsP2τi (dotted black). P0 : (σ0 =)β (k) 0 · · · β (k) s · · · α (k−1) t · · · α (k−1) r+1 (= τ0) · · · τ (k−1) i σ (k) j · · · P1 P2 σjP1αtP0βsP2τi [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: P1 (red) is a V-trajectory from σj to σ0 which meets P0 at αt . P2 (blue) is a W￾trajectory from σ0 to τi which leaves P0 at βs. For s < t, φ((P1, P2)) = σjP1αtP0βsP2τi (dotted black). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators.

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.

Referee Report

0 major / 3 minor

Summary. The paper claims that, for an admissible critical pair in a discrete Morse function on a simplicial complex, there is an explicit combinatorial formula expressing the modified boundary operators after cancellation directly in terms of the original boundary operators; the same update is realized by a finite sequence of elementary row operations on the incidence matrices. An analogous formula is derived for the coboundary operators. The derivation begins from the definition of the gradient vector field, isolates the unique reversing path, and shows that all incidence changes are captured by the stated algebraic relation, thereby avoiding re-enumeration of gradient trajectories.

Significance. If the derivation holds, the result supplies a direct algebraic update rule for the chain complex after an admissible cancellation. This removes the need to recompute gradient paths and therefore offers a practical efficiency gain for iterative simplification steps in discrete Morse homology computations. The construction is parameter-free once the original operators and the cancelled pair are given, and it respects the standard admissibility hypothesis already present in the literature.

minor comments (3)
  1. [§2] §2, Definition 2.3: the notation distinguishing the original boundary operator ∂ from the updated operator ∂' is introduced only after the main formula is stated; moving the definition earlier would improve readability of the subsequent derivation.
  2. [Theorem 3.4] Theorem 3.4: the statement that the row-operation sequence is 'equivalent' to the combinatorial formula would benefit from an explicit remark that both produce identical matrices (rather than merely homotopic complexes).
  3. [§4] The paper does not include a small worked example (e.g., a 2-complex with a single cancellable pair) that would allow immediate verification of the formula; adding one in §4 would strengthen the exposition without altering the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation of minor revision. The referee's summary accurately captures the main results concerning the explicit formulas for the updated boundary and coboundary operators after admissible critical pair cancellation.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct combinatorial derivation of an explicit algebraic formula relating the modified boundary (and coboundary) operators to the original ones after admissible cancellation of a critical pair. The derivation begins from the standard definition of a discrete gradient vector field, isolates the unique reversing path, and obtains the incidence-matrix update either by explicit path counting or by elementary row operations. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from the authors' prior work; the admissibility condition is the standard prerequisite already present in the literature. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard combinatorial properties of discrete Morse functions and chain complexes; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Existence and basic properties of gradient vector fields and critical simplices on simplicial complexes
    Invoked implicitly when discussing cancellation and its effect on boundary operators.

pith-pipeline@v0.9.0 · 5686 in / 1193 out tokens · 48686 ms · 2026-05-23T03:43:42.130851+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Discrete & Computational Geo metry 57, 824–853 (2017), https://doi.org/10.1007/s00454-017-9860-4

    Adiprasito, K.A., Benedetti, B., Lutz, F.H.: Extremal examples of c ollapsible complexes and random discrete Morse theory. Discrete & Computational Geo metry 57, 824–853 (2017), https://doi.org/10.1007/s00454-017-9860-4

    work page doi:10.1007/s00454-017-9860-4 2017

  2. [2]

    Journal of Applied and Computational Topology (2023), https://doi.org/10.1007/s41468-023-00139-4

    Benedetti, B., Lai, C., Lofano, D., Lutz, F.H.: Random simple- homotopy theory. Journal of Applied and Computational Topology (2023), https://doi.org/10.1007/s41468-023-00139-4

    work page doi:10.1007/s41468-023-00139-4 2023

  3. [3]

    Experimental Mathematics 23(1), 66–94 (2014), https://doi.org/10.1080/10586458.2013.865281

    Benedetti, B., Lutz, F.H.: Random discrete Morse theory and a ne w li- brary of triangulations. Experimental Mathematics 23(1), 66–94 (2014), https://doi.org/10.1080/10586458.2013.865281

    work page doi:10.1080/10586458.2013.865281 2014

  4. [4]

    Discrete Mathematics 217(1), 101–113 (2000), https://doi.org/10.1016/S0012-365X(99)00258-7

    Chari, M.K.: On discrete Morse functions and combinato- rial decompositions. Discrete Mathematics 217(1), 101–113 (2000), https://doi.org/10.1016/S0012-365X(99)00258-7

    work page doi:10.1016/s0012-365x(99)00258-7 2000

  5. [5]

    Computational Geometry 6(2), 85–98 (1996), https://doi.org/10.1016/0925-7721(95)00015-1

    Eˇ gecioˇ glu, ¨O., Gonzalez, T.F.: A computationally intractable problem on simplicial complexes. Computational Geometry 6(2), 85–98 (1996), https://doi.org/10.1016/0925-7721(95)00015-1

    work page doi:10.1016/0925-7721(95)00015-1 1996

  6. [6]

    Advances in Mathe matics 134(1), 90–145 (1998), https://doi.org/10.1006/aima.1997.1650

    Forman, R.: Morse theory for cell complexes. Advances in Mathe matics 134(1), 90–145 (1998), https://doi.org/10.1006/aima.1997.1650

    work page doi:10.1006/aima.1997.1650 1998

  7. [7]

    Tra nsac- tions of the American Mathematical Society 354(12), 5063–5085 (2002), https://doi.org/10.1090/S0002-9947-02-03041-6 17

    Forman, R.: Discrete morse theory and the cohomology ring. Tra nsac- tions of the American Mathematical Society 354(12), 5063–5085 (2002), https://doi.org/10.1090/S0002-9947-02-03041-6 17

    work page doi:10.1090/s0002-9947-02-03041-6 2002

  8. [8]

    S´ eminaire L otharingien de Combina- toire [electronic only] 48, B48c–35 (2002)

    Forman, R.: A user’s guide to discrete Morse theory. S´ eminaire L otharingien de Combina- toire [electronic only] 48, B48c–35 (2002)

    work page 2002

  9. [9]

    Annali della Scuola Normale Superiore di Pisa-Classe di Scien ze 9(2), 229–252 (2010), https://doi.org/10.2422/2036-2145.2010.2.01

    Gallais, E.: Combinatorial realization of the Thom–Smale complex via d iscrete Morse theory. Annali della Scuola Normale Superiore di Pisa-Classe di Scien ze 9(2), 229–252 (2010), https://doi.org/10.2422/2036-2145.2010.2.01

    work page doi:10.2422/2036-2145.2010.2.01 2010

  10. [10]

    Advances in A pplied Math- ematics 35(3), 294–322 (Sep 2005)

    Hersh, P.: On optimizing discrete Morse functions. Advances in A pplied Math- ematics 35(3), 294–322 (Sep 2005). https://doi.org/10.1016/j.aam.2005.04 .001, http://dx.doi.org/10.1016/j.aam.2005.04.001

    work page doi:10.1016/j.aam.2005.04 2005

  11. [11]

    Electronic Notes in Discrete Mathematics 17, 191–195 (2004), https://doi.org/10.1016/j.endm.2004.03.038

    Joswig, M., Pfetsch, M.E.: Computing optimal discrete Morse fun c- tions. Electronic Notes in Discrete Mathematics 17, 191–195 (2004), https://doi.org/10.1016/j.endm.2004.03.038

    work page doi:10.1016/j.endm.2004.03.038 2004

  12. [12]

    World Scien tific Publishing (Jul 2015), https://doi.org/10.1142/9360

    Knudson, K.P.: Morse theory: Smooth and discrete. World Scien tific Publishing (Jul 2015), https://doi.org/10.1142/9360

    work page doi:10.1142/9360 2015

  13. [13]

    Graduate studies in mathematics, American Mathematical Society, Providenc e, RI (Sep 2020)

    Kozlov, D.N.: Organized collapse: An introduction to discrete Mor se theory. Graduate studies in mathematics, American Mathematical Society, Providenc e, RI (Sep 2020)

    work page 2020

  14. [14]

    Lewiner, T., Lopes, H., Tavares, G.: Toward optimality in discrete morse theory. Exp. Math. 12(3), 271–285 (Jan 2003), https://doi.org/10.1080/10586458.2003.10504498

    work page doi:10.1080/10586458.2003.10504498 2003

  15. [15]

    Discrete Mathematics & Theoretical Co mputer Science vol

    Mondal, A., Mukherjee, S., Saha, K.: Topology of matching comple xes of complete graphs via discrete Morse theory. Discrete Mathematics & Theoretical Co mputer Science vol. 26:3 (2024), https://doi.org/10.46298/dmtcs.12887

    work page doi:10.46298/dmtcs.12887 2024

  16. [16]

    CRC Press, Lond on, England (Jun 2019), https://doi.org/10.1201/9780429493911

    Munkres, J.R.: Elements of algebraic topology. CRC Press, Lond on, England (Jun 2019), https://doi.org/10.1201/9780429493911

    work page doi:10.1201/9780429493911 2019

  17. [17]

    Scoville, N.A.: Discrete Morse theory. Student mathematical libr ary, American Mathemat- ical Society, Providence, RI (Dec 2019) Appendix A An example of simultaneous cancellation In [ 15, Section 4], the (discrete) Morse homology groups of the matching complex of the complete graph of order 7 are computed with respect to a ‘near optimal’ grad ient vector...

    work page 2019