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arxiv: 2508.17629 · v2 · pith:OYF2PVFMnew · submitted 2025-08-25 · 🧮 math.AT

On the complexity of parametrized motion planning algorithms

Navnath Daundkar , Ekansh Jauhari This is my paper

Pith reviewed 2026-05-25 08:12 UTC · model grok-4.3

classification 🧮 math.AT
keywords parametrized topological complexityprobabilistic variantmotion planning algorithmsFadell-Neuwirth fibrationsreal projective spacesequivariant homotopy theorycohomologytopological robotics
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The pith

A probabilistic variant of the r-th sequential parametrized topological complexity lower-bounds the classical invariant and shows different behavior on real projective space bundles.

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

The paper introduces a probabilistic variant of the r-th sequential parametrized topological complexity. This new invariant provides a lower bound on the classical version and measures the difficulty of building permissive parametrized motion planning algorithms. Cohomology calculations show it behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles. Equivariant homotopy theory shows markedly different behavior on bundles with real projective space fibers and special orthogonal structure groups. The work also examines further properties and relations to other invariants in topological robotics.

Core claim

The authors define a probabilistic variant of the r-th sequential parametrized topological complexity that bounds the classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. Using cohomology, this variant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles. Using equivariant homotopy theory, it behaves wildly differently on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups.

What carries the argument

The probabilistic variant of the r-th sequential parametrized topological complexity, which lower-bounds the classical invariant and is analyzed via cohomology on some fibrations and equivariant homotopy theory on others.

If this is right

  • The variant supplies a concrete lower bound that can be used when estimating the complexity of permissive parametrized motion planning.
  • Results already known for the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles transfer directly to the probabilistic version.
  • The sharp differences on real projective space bundles indicate new distinctions in motion planning complexity for those spaces.
  • Relations to other topological robotics invariants allow cross-comparisons and further calculations.

Where Pith is reading between the lines

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

  • The invariant may help design more permissive motion planning algorithms in practical robotics settings.
  • Computations on further classes of bundles could expose additional patterns of agreement or difference.
  • The approach might connect to probabilistic techniques used in other parts of algebraic topology.
  • Extensions to non-sequential or higher-order versions of the invariant are conceivable.

Load-bearing premise

The probabilistic variant is well-defined as an invariant, the lower bound relation to the classical complexity holds, and the cohomology and equivariant homotopy computations on the listed bundles are accurate.

What would settle it

An explicit computation of both invariants on a real projective space bundle with special orthogonal structure group showing that the probabilistic value fails to lower-bound the classical value.

read the original abstract

We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.

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 / 1 minor

Summary. The paper introduces a probabilistic variant of the r-th sequential parametrized topological complexity. It proves that this variant lower-bounds the classical invariant and measures the difficulty of constructing permissive parametrized motion planning algorithms. Using cohomology, the paper shows that the new invariant behaves similarly to the classical one on Fadell-Neuwirth fibrations and oriented sphere bundles; using equivariant homotopy theory, it shows that the behavior differs markedly on bundles with real projective space fibers and special orthogonal structure groups. The paper also examines additional properties of the invariant and its relations to other topological robotics invariants.

Significance. If the definitions, lower-bound relation, and bundle computations hold, the work supplies a new lower-bound tool for parametrized topological complexity that distinguishes geometric settings in a way the classical invariant does not, thereby refining the analysis of motion-planning algorithms on fibrations.

minor comments (1)
  1. The abstract asserts the existence of cohomology and equivariant homotopy computations but supplies no explicit definitions, notation, or sample calculations; this makes the central claims difficult to assess from the abstract alone even though the full manuscript is stated to be available.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential significance of the probabilistic variant as a distinguishing lower-bound tool. The recommendation is listed as uncertain, but the report contains no specific major comments or points of concern. We therefore have no revisions or point-by-point responses to offer at this time.

Circularity Check

0 steps flagged

No significant circularity; new invariant defined and bounded via standard cohomology and equivariant homotopy computations

full rationale

The paper introduces a probabilistic variant of sequential parametrized topological complexity and proves it bounds the classical invariant from below, with explicit comparisons on specific bundles (Fadell-Neuwirth fibrations, sphere bundles, RP^n bundles with SO structure groups) via cohomology and equivariant homotopy theory. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described claims. The derivations rest on external algebraic topology machinery rather than reducing to the paper's own inputs by construction. The central claims are therefore independent of the new invariant's definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper's central addition is the definition of the new probabilistic invariant itself; all other content rests on standard algebraic topology.

axioms (1)
  • standard math Standard properties and exact sequences of cohomology and equivariant homotopy theory
    Invoked to establish the behaviors on the listed fibrations and bundles.
invented entities (1)
  • probabilistic variant of the r-th sequential parametrized topological complexity no independent evidence
    purpose: To provide a lower bound on the classical invariant and to measure difficulty of permissive motion planning
    Newly introduced object whose definition and properties constitute the paper's contribution.

pith-pipeline@v0.9.0 · 5625 in / 1250 out tokens · 38284 ms · 2026-05-25T08:12:03.200676+00:00 · methodology

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Forward citations

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 2 Pith papers · 2 internal anchors

  1. [1]

    Equivariant and invariant parametrized topological complexity

    Ramandeep Singh Arora and Navnath Daundkar. Equivariant and invariant parametrized topological complexity. Preprint, arXiv:2412.12921, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  2. [2]

    Rudyak, and Dai Tamaki

    Ibai Basabe, Jesús González, Yuli B. Rudyak, and Dai Tamaki. Higher topological complexity and its symmetrization. Algebr. Geom. Topol., 14(4):2103–2124, 2014

    work page 2014

  3. [3]

    Higher equivariant and invariant topological complexities

    Marzieh Bayeh and Soumen Sarkar. Higher equivariant and invariant topological complexities. J. Homo- topy Relat. Struct., 15(3-4):397–416, 2020. 29

    work page 2020

  4. [4]

    The category of a map and of a cohomology class

    Israel Berstein and Tudor Ganea. The category of a map and of a cohomology class. Fund. Math., 50:265–279, 1961/62

    work page 1961

  5. [5]

    Pavle V. M. Blagojević and Günter M. Ziegler. Convex equipartitions via equivariant obstruction theory. Israel J. Math., 200:49–77, 2014

    work page 2014

  6. [6]

    Cohen, Michael Farber, and Shmuel Weinberger

    Daniel C. Cohen, Michael Farber, and Shmuel Weinberger. Topology of parametrized motion planning algorithms. SIAM J. Appl. Algebra Geom., 5(2):229–249, 2021

    work page 2021

  7. [7]

    Cohen, Michael Farber, and Shmuel Weinberger

    Daniel C. Cohen, Michael Farber, and Shmuel Weinberger. Parametrized topological complexity of collision-free motion planning in the plane. Ann. Math. Artif. Intell., 90(10):999–1015, 2022

    work page 2022

  8. [8]

    Equivariant topological complexity

    Hellen Colman and Mark Grant. Equivariant topological complexity. Algebr. Geom. Topol., 12(4):2299– 2316, 2012

    work page 2012

  9. [9]

    Surveys Monogr

    Octva Cornea, Gregory Lupton, John Oprea, and Daniel Tanré.Lusternik–Schnirelmann Category, volume 103 of Math. Surveys Monogr. AMS, Providence, 2003

    work page 2003

  10. [10]

    Equivariant parametrized topological complexity

    Navnath Daundkar. Equivariant parametrized topological complexity. Proc. Roy. Soc. Edinburgh Sect. A, page 1–24, 2024

    work page 2024

  11. [11]

    Distributional topological complexity of groups.Preprint, arXiv:2404.03041, 2024

    Alexander Dranishnikov. Distributional topological complexity of groups.Preprint, arXiv:2404.03041, 2024

    work page arXiv 2024

  12. [12]

    Distributional topological complexity and LS-category

    Alexander Dranishnikov and Ekansh Jauhari. Distributional topological complexity and LS-category. In Michael Farber and Jesús González, editors,Topology and AI—topological aspects of algorithms for autonomous motion, volume 4 of EMS Ser. Ind. Appl. Math., pages 363–385. EMS Press, Berlin, 2024

    work page 2024

  13. [13]

    Fadell and Sufian Y

    Edward R. Fadell and Sufian Y. Husseini.Geometry and Topology of Configuration Spaces. Springer Monogr. Math. Springer-Verlag, Berlin, 2001

    work page 2001

  14. [14]

    Fadell and Lee Neuwirth

    Edward R. Fadell and Lee Neuwirth. Configuration spaces. Math. Scand., 10:111–118, 1962

    work page 1962

  15. [15]

    Topological complexity of motion planning

    Michael Farber. Topological complexity of motion planning. Discrete Comput. Geom., 29(2):211–221, 2003

    work page 2003

  16. [16]

    Topology of robot motion planning

    Michael Farber. Topology of robot motion planning. In Paul Biran et al., editor, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, volume 217 of NATO Sci. Ser. II Math. Phys. Chem., pages 185–230. Springer, Dordrecht, 2006

    work page 2006

  17. [17]

    Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles

    Michael Farber, Mark Grant, and Sergey Yuzvinsky. Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles. In Michael Farber et. al., editor, Topology and Robotics, volume 438 of Contemp. Math., pages 75–83. AMS, Providence, 2007

    work page 2007

  18. [18]

    Sequential parametrized topological complexity and related invariants

    Michael Farber and John Oprea. Sequential parametrized topological complexity and related invariants. Algebr. Geom. Topol., 24(3):1755–1780, 2024

    work page 2024

  19. [19]

    Sequential parametrized motion planning and its complexity

    Michael Farber and Amit Kumar Paul. Sequential parametrized motion planning and its complexity. Topology Appl., 321:Paper No. 108256, pp. 23, 2022

    work page 2022

  20. [20]

    Sequential parametrized motion planning and its complexity, II

    Michael Farber and Amit Kumar Paul. Sequential parametrized motion planning and its complexity, II. Topology Appl., 331:Paper No. 108490, pp 9, 2023

    work page 2023

  21. [21]

    Sequential parametrized topological complexity of sphere bundles

    Michael Farber and Amit Kumar Paul. Sequential parametrized topological complexity of sphere bundles. Eur. J. Math., 10(4):Paper No. 53, pp 29, 2024

    work page 2024

  22. [22]

    Topological robotics: motion planning in projective spaces

    Michael Farber, Serge Tabachnikov, and Sergey Yuzvinsky. Topological robotics: motion planning in projective spaces. Int. Math. Res. Not., (34):1853–1870, 2003

    work page 2003

  23. [23]

    Parametrized motion planning and topological complexity

    Michael Farber and Shmuel Weinberger. Parametrized motion planning and topological complexity. In Steven M. LaValleet al., editor, Algorithmic Foundations of Robotics XV, volume 25 of Springer Proc. Adv. Robot., pages 1–17. Springer, Cham, 2023

    work page 2023

  24. [24]

    Parametrized topological complexity of sphere bundles.Topol

    Michael Farber and Shmuel Weinberger. Parametrized topological complexity of sphere bundles.Topol. Methods Nonlinear Anal., 61(1):161–177, 2023

    work page 2023

  25. [25]

    García-Calcines

    José M. García-Calcines. A note on covers defining relative and sectional categories. Topology Appl., 265:Paper No. 106810, pp. 14, 2019

    work page 2019

  26. [26]

    García-Calcines

    José M. García-Calcines. Formal aspects of parametrized topological complexity and its pointed version. J. Topol. Anal., 15(4):1129–1148, 2023

    work page 2023

  27. [27]

    Sequential motion planning of non-colliding particles in Euclidean spaces

    Jesús González and Mark Grant. Sequential motion planning of non-colliding particles in Euclidean spaces. Proc. Amer. Math. Soc., 143(10):4503–4512, 2015

    work page 2015

  28. [28]

    Topological complexity of motion planning in projective product spaces

    Jesús González, Mark Grant, Enrique Torres-Giese, and Miguel Xicoténcatl. Topological complexity of motion planning in projective product spaces. Algebr. Geom. Topol., 13(2):1027–1047, 2013

    work page 2013

  29. [29]

    Algebraic Topology

    Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002

    work page 2002

  30. [30]

    Ioan M. James. On category, in the sense of Lusternik–Schnirelmann. Topology, 17:331–348, 1978. 30

    work page 1978

  31. [31]

    LS-category and sequential topological complexity of symmetric products

    Ekansh Jauhari. LS-category and sequential topological complexity of symmetric products. Preprint, arXiv:2503.04532, 2025

    work page arXiv 2025

  32. [32]

    On sequential versions of distributional topological complexity.Topology Appl., 363:Paper No

    Ekansh Jauhari. On sequential versions of distributional topological complexity.Topology Appl., 363:Paper No. 109271, pp. 28, 2025

    work page 2025

  33. [33]

    On distributional one-category, diagonal distributional complexity, and related invariants

    Ekansh Jauhari and John Oprea. On distributional one-category, diagonal distributional complexity, and related invariants. Preprint, arXiv:2508.06973, 2025

    work page arXiv 2025

  34. [34]

    Symmetric joins and weighted barycenters.Adv

    Sadok Kallel and Rym Karoui. Symmetric joins and weighted barycenters.Adv. Nonlinear Stud., 11(1):117– 143, 2011

    work page 2011

  35. [35]

    Analog category and complexity.SIAM J

    Ben Knudsen and Shmuel Weinberger. Analog category and complexity.SIAM J. Appl. Algebra Geom., 8(3):713–732, 2024

    work page 2024

  36. [36]

    On the analog category of finite groups

    Ben Knudsen and Shmuel Weinberger. Analog category of finite groups.Algebr. Geom. Topol. (to appear), arXiv:2411.14958, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  37. [37]

    Pull-backs in homotopy theory

    Michael Mather. Pull-backs in homotopy theory. Canad. J. Math., 28(2):225–263, 1976

    work page 1976

  38. [38]

    Prokhorov

    Yuri V. Prokhorov. Convergence of random processes and limit theorems in probability theory.Theory Probab. Appl., 1:157–214, 1956

    work page 1956

  39. [39]

    Yuli B. Rudyak. On higher analogs of topological complexity. Topology Appl., 157(5):916–920, 2010

    work page 2010

  40. [40]

    Albert S. Schwarz. The genus of a fiber space. In Eleven papers on topology and algebra, volume 55 of Amer. Math. Soc. Transl. Ser. 2, pages 49–140. AMS, Providence, 1966

    work page 1966

  41. [41]

    Edwin H. Spanier. Algebraic Topology. McGraw-Hill Inc., New Y ork, 1966

    work page 1966

  42. [42]

    On the Lusternik–Schnirelmann category of Peano continua

    Tulsi Srinivasan. On the Lusternik–Schnirelmann category of Peano continua. Topology Appl., 160(13):1742–1749, 2013

    work page 2013

  43. [43]

    The Lusternik–Schnirelmann category of metric spaces

    Tulsi Srinivasan. The Lusternik–Schnirelmann category of metric spaces. Topology Appl., 167:87–95, 2014

    work page 2014

  44. [44]

    The Topology of Fibre Bundles, volume 14 of Princeton Math

    Norman Steenrod. The Topology of Fibre Bundles, volume 14 of Princeton Math. Ser. Princeton University Press, Princeton, 1951

    work page 1951

  45. [45]

    On pontrjagin classes and homotopy types of manifolds

    Itiro Tamura. On pontrjagin classes and homotopy types of manifolds. J. Math. Soc. Japan, 9(2):250–262, 1957

    work page 1957

  46. [46]

    John J. Walsh. Dimension, cohomological dimension, and cell-like mappings. In Sibe Mardešić and Jack Segal, editors, Shape Theory and Geometric Topology, volume 870 of Lecture Notes in Math., pages 105–118. Springer, Berlin, 1981

    work page 1981

  47. [47]

    Zarichny˘ı

    Michael M. Zarichny˘ı. Free topological groups of absolute neighborhood retracts and infinite-dimensional manifolds. Soviet Math. Dokl., 26:367–371, 1982. NAVNATH DAUNDKAR, DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH PUNE, INDIA. Email address: navnath.daundkar@acads.iiserpune.ac.in EKANSH JAUHARI, DEPARTMENT OF MATHEMATI...

    work page 1982