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arxiv: 2604.20004 · v1 · submitted 2026-04-21 · 🧮 math.AT · cs.CG· math.CT

Recognition: unknown

A continuum of K\"unneth theorems for persistence modules

Nikola Mili\'cevi\'c

Pith reviewed 2026-05-10 00:22 UTC · model grok-4.3

classification 🧮 math.AT cs.CGmath.CT
keywords persistence modulesKünneth theoremtensor productinternal homuniversal coefficient theoremposetsmulti-parameter persistencefiltered complexes
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0 comments X

The pith

For every order-preserving map on a poset, the induced tensor product of persistence modules gives a Künneth short exact sequence.

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

The paper introduces tensor products of persistence modules indexed by a poset P that are parametrized by order-preserving maps φ from P times P to P. It shows each such tensor product has an internal hom adjoint and that both produce Künneth short exact sequences when applied to chain complexes of persistence modules. These sequences hold in one-parameter and multi-parameter settings and specialize to universal coefficient theorems. The results apply to filtered CW complexes and enable faster persistent homology computations on product metric spaces with p-norm distances.

Core claim

For an arbitrary order-preserving map φ: P × P → P, the tensor product ⊗_φ on persistence modules indexed by P admits a right adjoint Hom^φ. Moreover, ⊗_φ induces a Künneth short exact sequence for the homology of tensor products of chain complexes of persistence modules, and Hom^φ induces a dual Künneth sequence in cohomology. Special cases recover universal coefficient theorems, and the constructions extend to filtered CW complexes.

What carries the argument

The φ-parametrized tensor product ⊗_φ of persistence modules and its right adjoint internal hom Hom^φ, which together carry the homological algebra needed for the Künneth sequences.

If this is right

  • Special cases of the sequences become universal coefficient theorems.
  • The theorems apply to chain complexes arising from filtered CW complexes.
  • They allow computation of persistent Borel-Moore homology for filtrations of non-compact spaces.
  • For p-quasinorms on the nonnegative reals, the sequences speed up persistent homology calculations in product metric spaces.

Where Pith is reading between the lines

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

  • The generality to arbitrary posets suggests the theorems apply uniformly across one-parameter and multi-parameter persistence without separate cases.
  • Explicit derived functors for interval modules may support direct calculations in specific filtrations.

Load-bearing premise

The constructions and exactness properties hold when φ is any order-preserving map from P times P to P, for arbitrary posets P.

What would settle it

A concrete poset P, order-preserving φ, and chain complex of persistence modules for which the associated Künneth sequence fails to be exact.

Figures

Figures reproduced from arXiv: 2604.20004 by Nikola Mili\'cevi\'c.

Figure 1
Figure 1. Figure 1: FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIGURE 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIGURE 3 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIGURE 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIGURE 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIGURE 6 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIGURE 7 [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIGURE 8 [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIGURE 9 [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIGURE 10 [PITH_FULL_IMAGE:figures/full_fig_p040_10.png] view at source ↗
read the original abstract

We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For a poset $P$ and an order preserving map $\varphi:P\times P\to P$, we introduce a novel tensor product of persistence modules indexed by $P$, $\otimes_{\varphi}$. We prove that each $\otimes_{\varphi}$ has a right adjoint, $\mathbf{Hom}^{\varphi}$, the internal hom of persistence modules that also depends on $\varphi$. We prove that every $\otimes_{\varphi}$ yields a K\"unneth short exact sequence of chain complexes of persistence modules. Dually, the $\mathbf{Hom}^{\varphi}$ also has an associated K\"unneth short exact sequence in cohomology. As special cases both of these short exact sequences yield Universal Coefficient Theorems. We show how to apply these to chain complexes of persistence modules arising from filtered CW complexes. For the special case of $P=\mathbb{R}_+$, the $p$-quasinorms for each $p\in (0,\infty]$ yield a distinct $\otimes_{\ell^p_c}$ and its adjoint $\mathbf{Hom}^{\ell^p_c}$. We compute their derived functors, $\mathbf{Tor}^{\ell^p_c}$ and $\mathbf{Ext}_{\ell^p_c}$ explicitly for interval modules. We show that the Universal Coefficient Theorem developed can be used to compute persistent Borel-Moore homology of a filtration of non-compact spaces. Finally, we show that for every $p\in [1,\infty]$ the associated K\"unneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space $(X\times Y,d^p)$ with the distance $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.

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

2 major / 2 minor

Summary. The paper introduces, for a poset P and any order-preserving map φ: P×P→P, a tensor product ⊗_φ on P-indexed persistence modules together with its right adjoint Hom^φ. It proves that ⊗_φ induces a Künneth short exact sequence on chain complexes of persistence modules and that Hom^φ induces a dual Künneth short exact sequence in cohomology; both sequences specialize to universal coefficient theorems. For P=ℝ₊ the p-quasinorms yield concrete functors ⊗_{ℓ^p_c} whose derived functors Tor and Ext are computed explicitly on interval modules. The results are applied to filtered CW complexes, persistent Borel-Moore homology of non-compact spaces, and accelerated persistent-homology computation on product metric spaces (X×Y,d^p).

Significance. If the stated adjunctions and exact sequences hold in the claimed generality, the work supplies a flexible, parameterized extension of homological algebra for persistence modules that unifies and generalizes existing Künneth and UCT results while furnishing explicit computational tools for multi-parameter settings and product filtrations. The explicit Tor/Ext formulas for interval modules and the demonstrated speed-up for persistent homology in product spaces are concrete strengths.

major comments (2)
  1. [§3.2] §3.2 (definition of ⊗_φ and proof of adjunction): the construction of ⊗_φ from an arbitrary order-preserving φ is presented as yielding a bifunctor on persistence modules, but the verification that the resulting object satisfies the persistence-module axioms (monotonicity and the required colimit preservation) is only sketched; a fully expanded check for general posets P (including multi-parameter cases) is needed to support the subsequent claims that the functor is exact and admits the stated right adjoint.
  2. [§4] §4 (Künneth short exact sequence for ⊗_φ): the derivation of the short exact sequence relies on a spectral-sequence argument or direct diagram chase that assumes the tensor product preserves exactness in each variable; for non-monoidal φ this preservation is not automatic on general posets, and the manuscript does not isolate the precise hypothesis on φ or on the chain complexes that guarantees the sequence remains short exact.
minor comments (2)
  1. [§5] Notation for the p-quasinorm functors ⊗_{ℓ^p_c} is introduced in §5 without an explicit comparison table relating the different p-values to the classical tensor product (p=1) and the sup-norm case (p=∞); adding such a table would improve readability.
  2. [§6.2] The application to persistent Borel-Moore homology in §6.2 cites filtered CW complexes but does not include a small explicit example (e.g., an infinite ray or half-plane) that would illustrate how the new UCT recovers known computations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested clarifications and expansions.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (definition of ⊗_φ and proof of adjunction): the construction of ⊗_φ from an arbitrary order-preserving φ is presented as yielding a bifunctor on persistence modules, but the verification that the resulting object satisfies the persistence-module axioms (monotonicity and the required colimit preservation) is only sketched; a fully expanded check for general posets P (including multi-parameter cases) is needed to support the subsequent claims that the functor is exact and admits the stated right adjoint.

    Authors: We agree that the verification of the persistence-module axioms for ⊗_φ in §3.2 was presented concisely. In the revised manuscript we will expand this section with a complete, self-contained argument that works for an arbitrary poset P. The expanded proof will explicitly verify monotonicity of the structure maps and preservation of the relevant colimits (including the case when P is a product poset arising in multi-parameter persistence). This expansion will also make transparent why ⊗_φ is exact in each variable and why the adjunction with Hom^φ holds without further restrictions on φ. revision: yes

  2. Referee: [§4] §4 (Künneth short exact sequence for ⊗_φ): the derivation of the short exact sequence relies on a spectral-sequence argument or direct diagram chase that assumes the tensor product preserves exactness in each variable; for non-monoidal φ this preservation is not automatic on general posets, and the manuscript does not isolate the precise hypothesis on φ or on the chain complexes that guarantees the sequence remains short exact.

    Authors: We acknowledge that the proof of the Künneth short exact sequence in §4 relies on exactness of ⊗_φ in each variable without isolating this property as a separate lemma. While the construction via colimits over the poset ensures exactness when the coefficient field is a field (standard in persistence), we will add an explicit lemma in the revision that states the precise conditions under which ⊗_φ preserves exact sequences. The lemma will cover arbitrary order-preserving φ and will be accompanied by a short diagram chase or reference to the spectral-sequence argument, thereby clarifying the hypotheses without narrowing the generality claimed for φ. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs ⊗_φ directly from an arbitrary order-preserving φ: P×P→P on a general poset P, then proves the existence of its right adjoint Hom^φ and the associated Künneth short exact sequences via standard categorical arguments (adjunctions and exactness properties of functors on persistence modules). These steps rely on first-principles category theory rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The special cases (p-quasinorms, UCTs, applications to filtered spaces) are derived as instances of the general theorems without reducing the central claims to their own inputs by construction. The development is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on the existence of a tensor product and internal hom defined via an arbitrary order-preserving map φ on a general poset, plus the standard axioms of abelian categories or module categories needed to obtain short exact sequences and derived functors. No free parameters are introduced; the new entities are the functors themselves.

axioms (1)
  • standard math Standard axioms of abelian categories and homological algebra (exactness properties, existence of derived functors Tor and Ext)
    Invoked to guarantee that the Künneth sequences and universal coefficient theorems exist once the tensor and hom are defined.
invented entities (2)
  • ⊗_φ tensor product of persistence modules no independent evidence
    purpose: To equip the category of persistence modules indexed by P with a tensor product that depends on the choice of order-preserving map φ
    Defined in the paper for each φ; no independent existence proof outside the construction is given.
  • Hom^φ internal hom no independent evidence
    purpose: Right adjoint to ⊗_φ, also depending on φ
    Introduced as the adjoint whose existence is proved; no external evidence supplied.

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Reference graph

Works this paper leans on

62 extracted references · 21 canonical work pages

  1. [1]

    The Vietoris–Rips complexes of a circle

    Michał Adamaszek and Henry Adams. “The Vietoris–Rips complexes of a circle”. Pacific Journal of Mathematics290.1 (2017), 1–40.ISSN: 0030-8730.DOI:10.2140/ pjm.2017.290.1

    2017

  2. [2]

    Vietoris–rips complexes of torus grids

    Henry Adams et al. “Vietoris–rips complexes of torus grids”. en.Mediterr. J. Math. 22.7 (Nov. 2025)

    2025

  3. [3]

    Mauricio Angel.2-Categorical Foundations for Multiparameter Persistence. 2025. arXiv:2508.02914 [math.AT]

    work page arXiv 2025

  4. [4]

    Nicolas Bonneel, Julien Rabin, Gabriel Peyr ´e, and Hanspeter Pfister

    Ulrich Bauer. “Ripser: efficient computation of Vietoris–Rips persistence barcodes”. Journal of Applied and Computational Topology5.3 (2021), 391–423.ISSN: 2367- 1734.DOI:10.1007/s41468-021-00071-5

    work page doi:10.1007/s41468-021-00071-5 2021

  5. [5]

    A derived isometry theorem for sheaves

    Nicolas Berkouk and Grégory Ginot. “A derived isometry theorem for sheaves”. en. Adv. Math. (N. Y.)394.108033 (Jan. 2022), p. 108033

    2022

  6. [6]

    Ephemeral persistence modules and distance comparison

    Nicolas Berkouk and François Petit. “Ephemeral persistence modules and distance comparison”. en.Algebr. Geom. Topol.21.1 (Feb. 2021), pp. 247–277

    2021

  7. [7]

    Nicolas Berkouk and Francois Petit.Projected distances for multi-parameter per- sistence modules. 2024. arXiv:2206.08818 [math.AT]

    work page arXiv 2024

  8. [8]

    Homological approxi- mations in persistence theory

    Benjamin Blanchette, Thomas Brüstle, and Eric J Hanson. “Homological approxi- mations in persistence theory”. en.Canad. J. Math.(Dec. 2022), pp. 1–38

    2022

  9. [9]

    Computing persistent homology with various coefficient fields in a single pass

    Jean-Daniel Boissonnat and Clément Maria. “Computing persistent homology with various coefficient fields in a single pass”. en.J. Appl. Comput. Topol.3.1-2 (June 2019), pp. 59–84

    2019

  10. [10]

    Homology theory for locally compact spaces

    A. Borel and J. C. Moore. “Homology theory for locally compact spaces.”Michi- gan Mathematical Journal7.2 (Jan. 1960).ISSN: 0026-2285.DOI:10 . 1307 / mmj / 1028998385

    1960

  11. [11]

    On the bottleneck stability of rank decompositions of multi-parameter persistence modules

    Magnus Bakke Botnan et al. “On the bottleneck stability of rank decompositions of multi-parameter persistence modules”. en.Adv. Math. (N. Y.)451.109780 (Aug. 2024), p. 109780

    2024

  12. [12]

    Bredon.Sheaf Theory

    Glen E. Bredon.Sheaf Theory. Springer New York, 1997.ISBN: 9781461206477.DOI: 10.1007/978-1-4612-0647-7

    work page doi:10.1007/978-1-4612-0647-7 1997

  13. [13]

    Statistical topological data analysis using persistence landscapes

    Peter Bubenik. “Statistical topological data analysis using persistence landscapes”.J. Mach. Learn. Res.16.1 (Jan. 2015), 77–102.ISSN: 1532-4435

    2015

  14. [14]

    Homological Algebra for Persistence Mod- ules

    Peter Bubenik and Nikola Milićević. “Homological Algebra for Persistence Mod- ules”.Foundations of Computational Mathematics21.5 (Jan. 2021), 1233–1278.ISSN: 1615-3383.DOI:10.1007/s10208-020-09482-9

    work page doi:10.1007/s10208-020-09482-9 2021

  15. [15]

    Categorification of persistent homology

    Peter Bubenik and Jonathan A Scott. “Categorification of persistent homology”. en. Discrete Comput. Geom.51.3 (Apr. 2014), pp. 600–627

    2014

  16. [16]

    Metrics for generalized persistence modules

    Peter Bubenik, Vin de Silva, and Jonathan Scott. “Metrics for generalized persistence modules”. en.Found. Comut. Math.15.6 (Dec. 2015), pp. 1501–1531

    2015

  17. [17]

    Topological spaces of persistence modules and their properties

    Peter Bubenik and Tane Vergili. “Topological spaces of persistence modules and their properties”. en.J. Appl. Comput. Topol.2.3-4 (Dec. 2018), pp. 233–269

    2018

  18. [18]

    American Mathematical Society, June 2001.ISBN: 9781470417949.DOI:10

    Dmitri Burago, Yuri Burago, and Sergei Ivanov.A Course in Metric Geometry. American Mathematical Society, June 2001.ISBN: 9781470417949.DOI:10 . 1090 / gsm/033. 49

    2001

  19. [19]

    Persistent homology of the sum met- ric

    Gunnar Carlsson and Benjamin Filippenko. “Persistent homology of the sum met- ric”. en.J. Pure Appl. Algebra224.5 (May 2020), p. 106244

    2020

  20. [20]

    Zigzag persistence

    Gunnar Carlsson and Vin de Silva. “Zigzag persistence”. en.Found. Comut. Math. 10.4 (Aug. 2010), pp. 367–405

    2010

  21. [21]

    The theory of multidimensional persis- tence

    Gunnar Carlsson and Afra Zomorodian. “The theory of multidimensional persis- tence”. en.Discrete Comput. Geom.42.1 (July 2009), pp. 71–93

    2009

  22. [22]

    Frederic Chazal et al.The structure and stability of persistence modules. en. 1st ed. SpringerBriefs in mathematics. Cham, Switzerland: Springer International Publish- ing, Oct. 2016

    2016

  23. [23]

    Persistence stability for geometric complexes

    Frédéric Chazal, Vin de Silva, and Steve Oudot. “Persistence stability for geometric complexes”.Geometriae Dedicata173.1 (Dec. 2013), 193–214.ISSN: 1572-9168.DOI: 10.1007/s10711-013-9937-z

    work page doi:10.1007/s10711-013-9937-z 2013

  24. [24]

    David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. “Stability of Persis- tence Diagrams”.Discrete & Computational Geometry37.1 (Dec. 2006), 103–120. ISSN: 1432-0444.DOI:10.1007/s00454-006-1276-5

    work page doi:10.1007/s00454-006-1276-5 2006

  25. [25]

    June 2022

    Marco Contessoto et al.Persistent Cup-Length. June 2022

    2022

  26. [26]

    Decomposition of pointwise finite-dimensional persis- tence modules

    William Crawley-Boevey. “Decomposition of pointwise finite-dimensional persis- tence modules”. en.J. Algebr. Appl.14.05 (June 2015), p. 1550066

    2015

  27. [27]

    Thesis (Ph.D.)–University of Pennsylvania

    Justin Michael Curry.Sheaves, cosheaves and applications. Thesis (Ph.D.)–University of Pennsylvania. ProQuest LLC, Ann Arbor, MI, 2014, p. 317.ISBN: 978-1303-96615-6

    2014

  28. [28]

    Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its appli- cations

    Tamal K Dey, Woojin Kim, and Facundo Mémoli. “Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its appli- cations”. en.Discrete Comput. Geom.71.1 (Jan. 2024), pp. 67–94

    2024

  29. [29]

    Homology Groups of Relations

    C. H. Dowker. “Homology Groups of Relations”.The Annals of Mathematics56.1 (1952), p. 84.ISSN: 0003-486X.DOI:10.2307/1969768

    work page doi:10.2307/1969768 1952

  30. [30]

    On the shape of a set of points in the plane

    H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. “On the shape of a set of points in the plane”.IEEE Transactions on Information Theory29.4 (1983), 551–559.ISSN: 1557-9654.DOI:10.1109/tit.1983.1056714

    work page doi:10.1109/tit.1983.1056714 1983

  31. [31]

    American Mathematical Society, 2010

    Herbert Edelsbrunner and John Harer.Computational Topology. American Math- ematical Society, Dec. 2009.ISBN: 9781470412081.DOI:10.1090/mbk/069

    work page doi:10.1090/mbk/069 2009

  32. [32]

    Persistence modules on commutative ladders of finite type

    Emerson G Escolar and Yasuaki Hiraoka. “Persistence modules on commutative ladders of finite type”. en.Discrete Comput. Geom.55.1 (Jan. 2016), pp. 100–157

    2016

  33. [33]

    Computation of gamma-linear projected barcodes for multiparameter persistence

    Alex Fernandes, Steve Oudot, and François Petit. “Computation of gamma-linear projected barcodes for multiparameter persistence”. en.J. Appl. Comput. Topol.9.2 (June 2025)

    2025

  34. [34]

    Perea.Künneth Formulae in Persistent Homology

    Hitesh Gakhar and Jose A. Perea.Künneth Formulae in Persistent Homology. 2019. arXiv:1910.05656 [math.AT]

    work page arXiv 2019

  35. [35]

    Global dimension of real-exponent polynomial rings

    Nathan Geist and Ezra Miller. “Global dimension of real-exponent polynomial rings”. en.Algebra Number Theory17.10 (Sept. 2023), pp. 1779–1788

    2023

  36. [36]

    Boris Goldfarb.Singular persistent homology with geometrically parallelizable computation. 2019. arXiv:1607.01257 [cs.CG]

    work page arXiv 2019

  37. [37]

    Florian Graf et al.The Flood Complex: Large-Scale Persistent Homology on Mil- lions of Points. 2026. arXiv:2509.22432 [cs.LG]

    work page arXiv 2026

  38. [38]

    Sur quelques points d’algèbre homologique, I

    Alexander Grothendieck. “Sur quelques points d’algèbre homologique, I”.Tohoku Math. J. (2)9.2 (Jan. 1957), pp. 119–221. 50

    1957

  39. [39]

    Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem

    Stéphane Guillermou and Pierre Schapira. “Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem”.Homological Mirror Symmetry and Trop- ical Geometry. Springer International Publishing, 2014, 43–85.ISBN: 9783319065144. DOI:10.1007/978-3-319-06514-4_3

    work page doi:10.1007/978-3-319-06514-4_3 2014

  40. [40]

    Stratifying multiparameter persistent homology

    Heather A Harrington et al. “Stratifying multiparameter persistent homology”. en. SIAM J. Appl. Algebr. Geom.3.3 (Jan. 2019), pp. 439–471

    2019

  41. [41]

    Manu Harsu and Eero Hyry.Ephemeral Modules and Scott Sheaves on a Con- tinuous Poset. 2024. arXiv:2411.16235 [math.AT].URL:https://arxiv.org/abs/ 2411.16235

    work page arXiv 2024

  42. [42]

    Les débuts de la théorie des faisceaux

    Masaki Kashiwara and Pierre Schapira.Sheaves on Manifolds. Springer Berlin Heidelberg, 1990.ISBN: 9783662026618.DOI:10.1007/978-3-662-02661-8

    work page doi:10.1007/978-3-662-02661-8 1990

  43. [43]

    Persistent homology and microlocal sheaf theory

    Masaki Kashiwara and Pierre Schapira. “Persistent homology and microlocal sheaf theory”. en.J. Appl. Comput. Topol.2.1-2 (Oct. 2018), pp. 83–113

    2018

  44. [44]

    Discrete Stratified Morse Theory: Algorithms and A User’s Guide

    Kevin Knudson and Bei Wang. “Discrete Stratified Morse Theory: Algorithms and A User’s Guide”.Discrete & Computational Geometry67.4 (Mar. 2022), 1023–1052. ISSN: 1432-0444.DOI:10.1007/s00454-022-00372-1

    work page doi:10.1007/s00454-022-00372-1 2022

  45. [45]

    Knudson and Nicholas A

    Kevin P. Knudson and Nicholas A. Scoville.Discrete Morse theory for open com- plexes. 2026. arXiv:2402.12116 [math.AT]

    work page arXiv 2026

  46. [46]

    The theory of the interleaving distance on multidimensional per- sistence modules

    Michael Lesnick. “The theory of the interleaving distance on multidimensional per- sistence modules”. en.Found. Comut. Math.15.3 (June 2015), pp. 613–650

    2015

  47. [47]

    Persistence Steenrod modules

    Umberto Lupo, Anibal M Medina-Mardones, and Guillaume Tauzin. “Persistence Steenrod modules”. en.J. Appl. Comput. Topol.6.4 (Dec. 2022), pp. 475–502

    2022

  48. [48]

    Medina-Mardones and Ling Zhou.Persistent cohomology operations and Gromov-Hausdorff estimates

    Anibal M. Medina-Mardones and Ling Zhou.Persistent cohomology operations and Gromov-Hausdorff estimates. 2025. arXiv:2503.17130 [math.AT]

    work page arXiv 2025

  49. [49]

    Persistent cup product structures and related invariants

    Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. “Persistent cup product structures and related invariants”. en.J. Appl. Comput. Topol.8.1 (Mar. 2024), pp. 93– 148

    2024

  50. [50]

    Stratifications of real vector spaces from constructible sheaves with conical microsupport

    Ezra Miller. “Stratifications of real vector spaces from constructible sheaves with conical microsupport”. en.J. Appl. Comput. Topol.7.3 (Sept. 2023), pp. 473–489

    2023

  51. [51]

    Homological algebra of modules over posets

    Ezra Miller. “Homological algebra of modules over posets”. en.SIAM J. Appl. Algebr. Geom.9.3 (Sept. 2025), pp. 483–524

    2025

  52. [52]

    Field choice problem in persistent homol- ogy

    Ippei Obayashi and Michio Yoshiwaki. “Field choice problem in persistent homol- ogy”. en.Discrete Comput. Geom.70.3 (Oct. 2023), pp. 645–670

    2023

  53. [53]

    Mathematical Surveys and Monographs

    Steve Y Oudot.Persistence theory: From quiver representations to data analysis. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, Sept. 2016

    2016

  54. [54]

    Persistence mod- ules with operators in Morse and floer theory

    Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević. “Persistence mod- ules with operators in Morse and floer theory”.Mosc. Math. J.17.4 (2017), pp. 757– 786

    2017

  55. [55]

    Computational tools in weighted per- sistent homology

    Shiquan Ren, Chengyuan Wu, and Jie Wu. “Computational tools in weighted per- sistent homology”. en.Chin. Ann. Math. Ser. B42.2 (Mar. 2021), pp. 237–258

    2021

  56. [56]

    Ana Romero et al.Defining and computing persistent Z-homology in the general case. 2014. arXiv:1403.7086 [cs.CG]

    work page arXiv 2014

  57. [57]

    Joseph J Rotman and Joseph J Rotman.An introduction to homological algebra. Vol. 2. Springer, 2009. 51

    2009

  58. [58]

    The persistent homology of a sampled map: from a viewpoint of quiver representations

    Hiroshi Takeuchi. “The persistent homology of a sampled map: from a viewpoint of quiver representations”. en.J. Appl. Comput. Topol.5.2 (June 2021), pp. 179–213

    2021

  59. [59]

    Microlocal Condition for Non-displaceability

    Dmitry Tamarkin. “Microlocal Condition for Non-displaceability”.Algebraic and Analytic Microlocal Analysis. Springer International Publishing, 2018, 99–223.ISBN: 9783030015886.DOI:10.1007/978-3-030-01588-6_3

    work page doi:10.1007/978-3-030-01588-6_3 2018

  60. [60]

    Distributing persistent homology via spectral sequences

    Álvaro Torras-Casas. “Distributing persistent homology via spectral sequences”. en. Discrete Comput. Geom.70.3 (Oct. 2023), pp. 580–619

    2023

  61. [61]

    Multiparameter Persistence Landscapes

    Oliver Vipond. “Multiparameter Persistence Landscapes”.Journal of Machine Learn- ing Research21.61 (2020), pp. 1–38

    2020

  62. [62]

    Computing persistent homology

    Afra Zomorodian and Gunnar Carlsson. “Computing persistent homology”. en.Dis- crete Comput. Geom.33.2 (Feb. 2005), pp. 249–274. 52

    2005