LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type

Koushik Brahma; Navnath Daundkar; Soumen Sarkar

arxiv: 2401.06680 · v2 · submitted 2024-01-12 · 🧮 math.AT

LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type

Koushik Brahma , Navnath Daundkar , Soumen Sarkar This is my paper

Pith reviewed 2026-05-24 04:09 UTC · model grok-4.3

classification 🧮 math.AT

keywords LS-categorytopological complexityreal torus manifoldsDold manifoldssmall coversequivariant LS-categorysymmetric topological complexitygeneralized real Bott manifolds

0 comments

The pith

Under certain hypotheses, the topological complexity of real torus manifolds of dimension n is either 2n or 2n+1.

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

The paper computes the LS-category of real torus manifolds, which generalize small covers, and of Dold manifolds of real torus type, which are fiber bundles over projective product spaces. It derives sharp bounds on topological complexity and proves that qualifying n-dimensional real torus manifolds have topological complexity exactly 2n or 2n+1. Tight bounds are given for generalized real Bott manifolds, with the gap often less than 5. The work also yields the Z2-equivariant LS-category for small covers whose fixed points are path-connected and exact symmetric topological complexity values for infinitely many of the manifolds considered.

Core claim

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. We compute the LS-category of these two types of manifolds and obtain sharp bounds on their topological complexities. We show that under certain hypotheses, the topological complexities of real torus manifolds of dimension n are either 2n or 2n+1. We figure out tight bounds for the topological complexity of generalized real Bott manifolds, and in many cases, the difference between these upper and lower bounds is less than 5. We compute the Z2-equivariant LS-category of small 0.

What carries the argument

The structural definitions of real torus manifolds as generalizations of small covers together with the fiber-bundle construction of Dold manifolds of real torus type, used under hypotheses such as path-connectedness of Z2-fixed point sets to bound LS-category and topological complexity.

If this is right

Topological complexity of qualifying real torus manifolds takes only the two values 2n and 2n+1.
Upper and lower bounds on topological complexity for generalized real Bott manifolds differ by less than 5 in many cases.
The Z2-equivariant LS-category equals a specific value for small covers whose Z2-fixed points form a path-connected set.
Symmetric topological complexity attains exact computed values for infinitely many of the manifolds studied.

Where Pith is reading between the lines

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

The same bundle and fixed-point techniques could be applied to compute topological complexity for other classes of manifolds equipped with involutions or torus actions.
The narrow gap between bounds for generalized real Bott manifolds indicates that additional algebraic invariants might close the gap to exact values.
Exact symmetric topological complexity results supply concrete test cases for comparing symmetric and ordinary topological complexity on the same spaces.

Load-bearing premise

The manifolds satisfy the structural definitions of real torus manifolds or Dold manifolds of real torus type and the stated hypotheses including path-connectedness of the Z2-fixed points hold.

What would settle it

A single real torus manifold of dimension n whose topological complexity is measured to be neither 2n nor 2n+1 would falsify the stated claim.

read the original abstract

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. In this paper, first, we compute the LS-category of these two types of manifolds and obtain sharp bounds on their topological complexities. We show that under certain hypotheses, the topological complexities of real torus manifolds of dimension $n$ are either $2n$ or $2n+1$.We figure out tight bounds for the topological complexity of generalized real Bott manifolds, and in many cases, the difference between these upper and lower bounds is less than 5. We compute the $\mathbb{Z}_2$-equivariant LS-category of small covers when the $\mathbb{Z}_2$-fixed points are path connected. In the end, we study the symmetric topological complexity of the above-mentioned manifolds and obtain exact values for infinitely many cases.

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 supplies new explicit calculations and bounds for LS-category and topological complexity on real torus manifolds and Dold manifolds of real torus type.

read the letter

The paper gives new explicit calculations for LS-category and topological complexity on real torus manifolds and Dold manifolds of real torus type. These are generalizations of small covers and Bott manifolds, and the authors claim sharp bounds like TC = 2n or 2n+1 under hypotheses, and small gaps for generalized real Bott manifolds. They also compute equivariant LS-category when Z2-fixed points are path connected and symmetric TC in infinitely many cases. What the paper does well is supply these concrete numbers and bounds for these specific classes, which extends the literature in a direct way. The soft spots are that everything is conditional on unspecified hypotheses, and without seeing the actual arguments it's impossible to tell if the computations are correct or if the conditions are too narrow to be interesting. The abstract is all we have here, so the soundness can't be assessed yet. This is for people in the subfield of LS-category and TC who want to see calculations on these manifold families. It could be useful for someone looking for examples or bounds in this area. It deserves a serious referee because the results are new and specific, even if the impact is limited. I would recommend sending it to peer review so the proofs can be checked.

Referee Report

1 major / 0 minor

Summary. The manuscript computes the LS-category of real torus manifolds and Dold manifolds of real torus type. It obtains sharp bounds on their topological complexities, showing that under certain hypotheses the topological complexities of real torus manifolds of dimension n are either 2n or 2n+1. It provides tight bounds for the topological complexity of generalized real Bott manifolds, with the difference between upper and lower bounds less than 5 in many cases. It computes the Z_2-equivariant LS-category of small covers when the Z_2-fixed points are path connected. Finally, it studies the symmetric topological complexity of these manifolds and obtains exact values for infinitely many cases.

Significance. If the results hold, they contribute concrete computations and bounds for LS-category and TC on these classes of manifolds, including equivariant and symmetric variants, with exact values in infinitely many cases. This adds to the body of explicit results in the area.

major comments (1)

The abstract asserts the existence of computations and bounds, but without the full proofs or methods it is impossible to check whether the derivations actually support the stated claims; major verification gap.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and positive summary of the manuscript. We address the single major comment below.

read point-by-point responses

Referee: The abstract asserts the existence of computations and bounds, but without the full proofs or methods it is impossible to check whether the derivations actually support the stated claims; major verification gap.

Authors: The full manuscript contains the complete proofs and methods. The LS-category computations for real torus manifolds appear in Section 3, those for Dold manifolds of real torus type in Section 4; the sharp TC bounds for real torus manifolds (showing values 2n or 2n+1 under the stated hypotheses) are proved in Section 5; tight bounds for generalized real Bott manifolds are derived in Section 6; the Z_2-equivariant LS-category result for small covers with path-connected fixed-point set is established in Section 7; and the exact symmetric TC values for infinitely many cases are obtained in Section 8. All derivations rely on standard tools from algebraic topology (cup-length lower bounds, sectional category upper bounds via fibrations, and equivariant methods) applied to the explicit cell structures and cohomology rings of these manifolds. The referee is referred to these sections for verification; we are prepared to expand any specific step if requested. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and context describe standard computations of LS-category and topological complexity for real torus manifolds and Dold manifolds, yielding bounds such as 2n or 2n+1 under stated hypotheses. No equations, definitions, or self-citations are supplied that reduce any claimed result to a fitted input, self-definition, or load-bearing prior work by the same authors. The derivation chain appears self-contained against external topological definitions and is not forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are visible from the abstract; the work rests on standard properties of LS-category and topological complexity.

axioms (1)

standard math Standard properties of LS-category and topological complexity as homotopy invariants
Computations rely on established definitions and properties of these invariants from algebraic topology.

pith-pipeline@v0.9.0 · 5697 in / 1088 out tokens · 29870 ms · 2026-05-24T04:09:40.126135+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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For a topolo gical space X, the topolog- ical complexity TC(X) is a numerical homotopy invariant

I/N.sc/T.sc/R.sc/O.sc/D.sc/U.sc/C.sc/T.sc/I.sc/O.sc/N.sc Farber [16] introduced the concept of topological complexity to study th e robot motion planning problem through the topological lens. For a topolo gical space X, the topolog- ical complexity TC(X) is a numerical homotopy invariant. Let X be a path-connected space and P X be the space of all paths i...

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LS-/C.sc/A.sc/T.sc/E.sc/G.sc/O.sc/R.sc/Y.sc /A.sc/N.sc/D.sc /T.sc/O.sc/P.sc/O.sc/L.sc/O.sc/G.sc/I.sc/C.sc/A.sc/L.sc /C.sc/O.sc/M.sc/P.sc/L.sc/E.sc/X.sc/I.sc/T.sc/Y.sc /O.sc/F.sc /R.sc/E.sc/A.sc/L.sc /T.sc/O.sc/R.sc/U.sc/S.sc /M.sc/A.sc/N.sc/I.sc/F.sc/O.sc/L.sc/D.sc/S.sc In this section, we recall the concepts of real torus manifol ds and generalized real ...

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(5) In particular, if n = 2 s, then 2s+1 ≤ TC(M) ≤ 2s+1 + 1

Then cat(M) = n + 1, and zclZ2(RP n) + 1 ≤ TC(M) ≤ 2n + 1. (5) In particular, if n = 2 s, then 2s+1 ≤ TC(M) ≤ 2s+1 + 1. Proof. W e can have M = M ′#RP n following the discussion in Subsection 1.11 in [ 11]. Then H ∗ (RP n; Z2) is a subring of H ∗ (M; Z2). Then n + 1 = clZ2(RP n) ≤ clZ2(M) ≤ n + 1, and the left inequality of ( 5) follows from the observati...

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corresponding to the characteristic function λ on P . W e recall the cohomology ring of a small cover over a product of simplice s from Proposition 2.11. Now we look at the generators Γj’s of the ideal I in the cohomology ring in Proposi- tion 2.11. Note that, Γ1 = x1x2 · · ·xn1y1 = yn1+1 1 , and Γj = xNj− 1+1xNj− 1+2 · · ·xNj yj = (yj + β j 11y1 + β j 21...

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Recall the height function f on a simple polytope P that was introduced in the proof of Theorem 3.1 of [ 11]

that q− 1(P ⟨ζ ′ ⟩) is path-connected. Recall the height function f on a simple polytope P that was introduced in the proof of Theorem 3.1 of [ 11]. One can orient the edges of polytope so that the f increases along them. This orientation makes the 1-skeleton of P into a directed graph. Theorem 3.2. Let M be an n-dimensional small cover such that M ⟨τ ⟩ i...

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In this section, we recall the basic terminologies related to symmetric topolo gical complexity and state some results that we are going to use further

S/Y.sc/M.sc/M.sc/E.sc/T.sc/R.sc/I.sc/C.sc /T.sc/O.sc/P.sc/O.sc/L.sc/O.sc/G.sc/I.sc/C.sc/A.sc/L.sc /C.sc/O.sc/M.sc/P.sc/L.sc/E.sc/X.sc/I.sc/T.sc/Y.sc /O.sc/F.sc /S.sc/M.sc/A.sc/L.sc/L.sc /C.sc/O.sc/V.sc/E.sc/R.sc/S.sc In [17], Farber and Grant studied the symmetric analogue of the moti on planning prob- lem and introduced the notion of symmetric topologica...

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[1] [1]

For a topolo gical space X, the topolog- ical complexity TC(X) is a numerical homotopy invariant

I/N.sc/T.sc/R.sc/O.sc/D.sc/U.sc/C.sc/T.sc/I.sc/O.sc/N.sc Farber [16] introduced the concept of topological complexity to study th e robot motion planning problem through the topological lens. For a topolo gical space X, the topolog- ical complexity TC(X) is a numerical homotopy invariant. Let X be a path-connected space and P X be the space of all paths i...

work page 2003

[2] [2]

W e give a nice presentation of the cohomology ring of generalized real Bott manifolds with Z2-coeﬃcients

LS-/C.sc/A.sc/T.sc/E.sc/G.sc/O.sc/R.sc/Y.sc /A.sc/N.sc/D.sc /T.sc/O.sc/P.sc/O.sc/L.sc/O.sc/G.sc/I.sc/C.sc/A.sc/L.sc /C.sc/O.sc/M.sc/P.sc/L.sc/E.sc/X.sc/I.sc/T.sc/Y.sc /O.sc/F.sc /R.sc/E.sc/A.sc/L.sc /T.sc/O.sc/R.sc/U.sc/S.sc /M.sc/A.sc/N.sc/I.sc/F.sc/O.sc/L.sc/D.sc/S.sc In this section, we recall the concepts of real torus manifol ds and generalized real ...

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(5) In particular, if n = 2 s, then 2s+1 ≤ TC(M) ≤ 2s+1 + 1

Then cat(M) = n + 1, and zclZ2(RP n) + 1 ≤ TC(M) ≤ 2n + 1. (5) In particular, if n = 2 s, then 2s+1 ≤ TC(M) ≤ 2s+1 + 1. Proof. W e can have M = M ′#RP n following the discussion in Subsection 1.11 in [ 11]. Then H ∗ (RP n; Z2) is a subring of H ∗ (M; Z2). Then n + 1 = clZ2(RP n) ≤ clZ2(M) ≤ n + 1, and the left inequality of ( 5) follows from the observati...

work page

[4] [4]

W e recall the cohomology ring of a small cover over a product of simplice s from Proposition 2.11

corresponding to the characteristic function λ on P . W e recall the cohomology ring of a small cover over a product of simplice s from Proposition 2.11. Now we look at the generators Γj’s of the ideal I in the cohomology ring in Proposi- tion 2.11. Note that, Γ1 = x1x2 · · ·xn1y1 = yn1+1 1 , and Γj = xNj− 1+1xNj− 1+2 · · ·xNj yj = (yj + β j 11y1 + β j 21...

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[5] [5]

Then, we compute some lower and upper bounds for the topological complexity of these manifolds

LS-/C.sc/A.sc/T.sc/E.sc/G.sc/O.sc/R.sc/Y.sc /A.sc/N.sc/D.sc /T.sc/O.sc/P.sc/O.sc/L.sc/O.sc/G.sc/I.sc/C.sc/A.sc/L.sc /C.sc/O.sc/M.sc/P.sc/L.sc/E.sc/X.sc/I.sc/T.sc/Y.sc /O.sc/F.sc D/O.sc/L.sc/D.sc /M.sc/A.sc/N.sc/I.sc/F.sc/O.sc/L.sc/D.sc/S.sc /O.sc/F.sc /R.sc/E.sc/A.sc/L.sc /T.sc/O.sc/R.sc/U.sc/S.sc /T.sc/Y.sc/P.sc/E.sc In this section, we recall Dold manif...

work page

[6] [6]

Recall the height function f on a simple polytope P that was introduced in the proof of Theorem 3.1 of [ 11]

that q− 1(P ⟨ζ ′ ⟩) is path-connected. Recall the height function f on a simple polytope P that was introduced in the proof of Theorem 3.1 of [ 11]. One can orient the edges of polytope so that the f increases along them. This orientation makes the 1-skeleton of P into a directed graph. Theorem 3.2. Let M be an n-dimensional small cover such that M ⟨τ ⟩ i...

work page

[7] [7]

In this section, we recall the basic terminologies related to symmetric topolo gical complexity and state some results that we are going to use further

S/Y.sc/M.sc/M.sc/E.sc/T.sc/R.sc/I.sc/C.sc /T.sc/O.sc/P.sc/O.sc/L.sc/O.sc/G.sc/I.sc/C.sc/A.sc/L.sc /C.sc/O.sc/M.sc/P.sc/L.sc/E.sc/X.sc/I.sc/T.sc/Y.sc /O.sc/F.sc /S.sc/M.sc/A.sc/L.sc/L.sc /C.sc/O.sc/V.sc/E.sc/R.sc/S.sc In [17], Farber and Grant studied the symmetric analogue of the moti on planning prob- lem and introduced the notion of symmetric topologica...

work page

[8] [8]

Moreover, it follows from [ 17, Example 7] that the symmetric topological complexity of a contractible space with more than two points is 2

that, in this case, TCS(Y ) = 1 and if Y contains more than one point, thenTCS(Y ) ≥ 2. Moreover, it follows from [ 17, Example 7] that the symmetric topological complexity of a contractible space with more than two points is 2. W e now describe a lower bound on the symmetric topological complexity given in [ 17]. Let NY be the sub-ring of H ∗ (Y )⊗ H ∗ (...

work page 2023

[9] [9]

Angel and H

A. Angel and H. Colman. Equivariant topological complex ities. T opological complexity and related topics, Contemp. Math, 702:1–15, 2018

work page 2018

[10] [10]

Bayeh and S

M. Bayeh and S. Sarkar. Some aspects of equivariant LS-ca tegory. T opology Appl., 196:133–154, 2015

work page 2015

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Berstein

I. Berstein. On the Lusternik-Schnirelmann category of Grassmannians. Math. Proc. Cambridge Philos. Soc., 79(1):129–134, 1976

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Brahma, B

K. Brahma, B. Naskar, S. Sarkar, and S. Sau. Various topol ogical complexities of small covers and real bott manifolds. arXiv preprint arXiv:2306.00916 , 2023. 14

work page arXiv 2023

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S. Choi, M. Masuda, and D. Y. Suh. Quasitoric manifolds ov er a product of simplices. Osaka J. Math. , 47(1):109–129, 2010

work page 2010

[14] [14]

Colman and M

H. Colman and M. Grant. Equivariant topological complex ity. Algebr. Geom. T opol., 12(4):2299–2316, 2012

work page 2012

[15] [15]

Cornea, G

O. Cornea, G. Lupton, J. Oprea, and D. Tanré. Lusternik-Schnirelmann category, volume 103 of Mathe- matical Surveys and Monographs . American Mathematical Society, Providence, RI, 2003

work page 2003

[16] [16]

Daundkar and S

N. Daundkar and S. Sarkar. LS-category and topological c omplexity of several families of ﬁbre bundles. Accepted for publication in homology homotopy and applicat ions, arXiv:2302.00468 , 2023

work page arXiv 2023

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D. M. Davis. Projective product spaces. J. T opol., 3(2):265–279, 2010

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M. W. Davis. Groups generated by reﬂections and aspheri cal manifolds not covered by Euclidean space. Ann. of Math. (2) , 117(2):293–324, 1983

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M. W. Davis and T. Januszkiewicz. Convex polytopes, Cox eter orbifolds and torus actions. Duke Math. J., 62(2):417–451, 1991

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A. Dold. Erzeugende der Thomschen Algebra N. Math. Z., 65:25–35, 1956

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Dranishnikov

A. Dranishnikov. On topological complexity of non-ori entable surfaces. T opology Appl., 232:61–69, 2017

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Dranishnikov

A. Dranishnikov. On topological complexity of hyperbo lic groups. Proc. Amer. Math. Soc., 148(10):4547– 4556, 2020

work page 2020

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Dsouza and V

R. Dsouza and V. Uma. Results on the topology of generali zed real Bott manifolds. Osaka J. Math. , 56(3):441–458, 2019

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