arxiv: 2605.08707 · v1 · submitted 2026-05-09 · 🧮 math.AT

Recognition: no theorem link

Homotopy exponents of polyhedral products

Briony Eldridge

Pith reviewed 2026-05-12 00:51 UTC · model grok-4.3

classification 🧮 math.AT

keywords polyhedral productshomotopy exponentsMoore's conjecturerational hyperbolicitysimplicial complexespolyhedral joinsalgebraic topology

0 comments

The pith

For polyhedral products built from finite spaces with torsion-free homology, rational hyperbolicity implies no homotopy exponent at any odd prime, and Moore's conjecture holds under an extra suspension condition.

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

The paper establishes that if a polyhedral product (CA, A)^K is rationally hyperbolic, with each A_i finite and having torsion-free homology, then the space has no homotopy exponent at odd primes. It shows this by relating the rational homotopy properties of the construction to its p-local behavior. Under the further assumption that the suspension of each A_i is a finite-type wedge of simply-connected spheres, the space satisfies Moore's conjecture. The work also supplies criteria ensuring that large families of polyhedral join products are rationally hyperbolic and lack homotopy exponents at all but finitely many primes. A sympathetic reader would care because these results connect rational invariants to the existence of exponents in a class of spaces that includes many toric and moment-angle complexes.

Core claim

For (CA, A)^K where each A_i is finite and has torsion-free homology, if (CA, A)^K is rationally hyperbolic, then it has no homotopy exponent at any odd prime. Under the additional hypothesis that Sigma A_i is homotopy equivalent to a finite-type wedge of simply-connected spheres, Moore's conjecture holds for (CA, A)^K.

What carries the argument

The polyhedral product (CA, A)^K constructed from a simplicial complex K and pairs (CA_i, A_i), which carries the argument by allowing control of rational hyperbolicity to imply the non-existence of odd-prime homotopy exponents.

If this is right

Rationally hyperbolic polyhedral products of this form have no homotopy exponent at odd primes.
Moore's conjecture is true for these polyhedral products when each suspended A_i is a finite-type wedge of simply-connected spheres.
Criteria on the simplicial complex and pairs yield polyhedral join products that are rationally hyperbolic and mod-p^r hyperbolic for all but finitely many primes.
Such spaces have no homotopy exponent at all but finitely many primes.

Where Pith is reading between the lines

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

The results suggest that rational hyperbolicity serves as a strong global obstruction to the existence of odd-primary homotopy exponents in these combinatorial constructions.
One could test the criteria by computing rational homotopy groups for explicit low-dimensional examples such as when K is a simplex or a disjoint union of edges.
The techniques may apply to related spaces like Davis-Januszkiewicz spaces or other polyhedral products arising in toric topology.

Load-bearing premise

Each A_i must be finite with torsion-free homology, the polyhedral product must be rationally hyperbolic, and for the Moore conjecture part the suspensions of the A_i must be finite-type wedges of simply-connected spheres.

What would settle it

A concrete counterexample would be any specific simplicial complex K and finite spaces A_i with torsion-free homology such that (CA, A)^K is rationally hyperbolic yet possesses a homotopy exponent at some odd prime, or fails Moore's conjecture when the suspension condition holds.

read the original abstract

We study Moore's conjecture and homotopy exponents for polyhedral products. For $(\underline{CA},\underline{A})^K$ where each $A_i$ is finite and has torsion-free homology, we prove that if $(\underline{CA},\underline{A})^K$ is rationally hyperbolic, then it has no homotopy exponent at any odd prime. Under the additional hypothesis $\Sigma A_i$ is homotopy equivalent to a finite-type wedge of simply-connected spheres, we show Moore's conjecture holds for $(\underline{CA},\underline{A})^K$. We also give criteria such that, for a large family of polyhedral join products, the associated polyhedral products are rationally hyperbolic, mod-$p^r$ hyperbolic for all but finitely many primes, and have no homotopy exponent at all but finitely many primes.

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 proves some conditional links between rational hyperbolicity and the absence of odd-prime homotopy exponents for certain polyhedral products, plus hyperbolicity criteria for joins.

read the letter

This paper proves that for polyhedral products (CA, A)^K where each A_i is finite with torsion-free homology, rational hyperbolicity implies no homotopy exponent at any odd prime. Under the extra assumption that the suspensions of the A_i are finite-type wedges of simply-connected spheres, Moore's conjecture holds for these spaces. It also gives criteria that make a large family of polyhedral join products rationally hyperbolic, mod-p^r hyperbolic for all but finitely many primes, and free of homotopy exponents at all but finitely many primes.

Referee Report

0 major / 2 minor

Summary. The manuscript studies Moore's conjecture and homotopy exponents for polyhedral products. For (CA, A)^K with each A_i finite and having torsion-free homology, it proves that rational hyperbolicity of the polyhedral product implies no homotopy exponent at any odd prime. Under the further hypothesis that ΣA_i is homotopy equivalent to a finite-type wedge of simply-connected spheres, Moore's conjecture holds for (CA, A)^K. Criteria are also given ensuring that large families of polyhedral join products yield spaces that are rationally hyperbolic, mod-p^r hyperbolic for all but finitely many primes, and have no homotopy exponent at all but finitely many primes.

Significance. If the derivations hold, the results advance the homotopy theory of polyhedral products by supplying explicit, hypothesis-driven conditions under which Moore's conjecture is verified and homotopy exponents are ruled out at odd primes. The conditional statements are cleanly stated, and the hyperbolicity criteria for join products supply a practical tool for generating examples, strengthening the paper's utility in algebraic topology.

minor comments (2)

The abstract and introduction would benefit from a brief reminder of the definition of the polyhedral product (CA, A)^K and the polyhedral join, as these notations are central but may not be immediately recalled by all readers.
In the statement of the main results, explicitly cross-reference the precise theorem numbers (e.g., Theorem 3.2 or 4.1) where the no-exponent claim and the Moore-conjecture claim are proved, to improve navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on Moore's conjecture and homotopy exponents for polyhedral products, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: theorems are conditional on external hypotheses and derived from standard tools

full rationale

The paper states conditional theorems for polyhedral products under explicit hypotheses (finite A_i with torsion-free homology, rational hyperbolicity, and suspension condition for Moore's conjecture). These are load-bearing but external to the derivation; the proofs invoke standard homotopy theory without reducing any claimed result to a fitted parameter, self-definition, or self-citation chain that loops back to the input. No equations or steps equate a prediction to its own construction by renaming or ansatz smuggling. The derivation remains self-contained against external benchmarks in algebraic topology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms and definitions from algebraic topology and rational homotopy theory; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)

standard math Standard properties of rational homotopy groups, hyperbolicity, and homotopy exponents in algebraic topology
Invoked to define the objects and state the theorems.
domain assumption Definitions of polyhedral products, polyhedral joins, and the given hypotheses on A_i
Core to the statements; these are standard in the area but load-bearing.

pith-pipeline@v0.9.0 · 5413 in / 1401 out tokens · 60266 ms · 2026-05-12T00:51:14.942833+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages

[1]

The homotopy type of the polyhedral product for shifted complexes , JOURNAL =

Grbi\'. The homotopy type of the polyhedral product for shifted complexes , JOURNAL =. 2013 , PAGES =. doi:10.1016/j.aim.2013.05.002 , URL =

work page doi:10.1016/j.aim.2013.05.002 2013
[2]

, TITLE =

Mather, M. , TITLE =. Canad. J. Math. , VOLUME =. 1976 , NUMBER =. doi:10.4153/CJM-1976-029-0 , URL =

work page doi:10.4153/cjm-1976-029-0 1976
[3]

Abramyan, S. A. and Panov, T. E. , TITLE =. Tr. Mat. Inst. Steklova , VOLUME =. 2019 , PAGES =. doi:10.4213/tm3995 , URL =

work page doi:10.4213/tm3995 2019
[5]

and Sun, Q

Hao, Y. and Sun, Q. and Theriault, S. , TITLE =. Math. Proc. Cambridge Philos. Soc. , VOLUME =. 2019 , NUMBER =. doi:10.1017/s0305004118000154 , URL =

work page doi:10.1017/s0305004118000154 2019
[6]

, TITLE =

McGavran, D. , TITLE =. Trans. Amer. Math. Soc. , VOLUME =. 1979 , PAGES =. doi:10.2307/1998691 , URL =

work page doi:10.2307/1998691 1979
[7]

and Theriault, S

Beben, P. and Theriault, S. , TITLE =. Adv. Math. , VOLUME =. 2014 , PAGES =. doi:10.1016/j.aim.2014.05.015 , URL =

work page doi:10.1016/j.aim.2014.05.015 2014
[8]

The homotopy type of the complement of a coordinate subspace arrangement , JOURNAL =

Grbi\'. The homotopy type of the complement of a coordinate subspace arrangement , JOURNAL =. 2007 , NUMBER =. doi:10.1016/j.top.2007.02.006 , URL =

work page doi:10.1016/j.top.2007.02.006 2007
[9]

and Kishimoto, D

Iriye, K. and Kishimoto, D. , TITLE =. J. Math. Soc. Japan , VOLUME =. 2020 , NUMBER =. doi:10.2969/jmsj/82708270 , URL =

work page doi:10.2969/jmsj/82708270 2020
[10]

Stanton , title =

L. Stanton , title =. Q. J. Math. , year =

work page
[11]

and Theriault, S

Panov, T. and Theriault, S. , TITLE =. Compos. Math. , VOLUME =. 2019 , NUMBER =. doi:10.1112/s0010437x18007613 , URL =

work page doi:10.1112/s0010437x18007613 2019
[12]

, TITLE =

Theriault, S. , TITLE =. Tr. Mat. Inst. Steklova , VOLUME =. 2022 , PAGES =. doi:10.4213/tm4284 , URL =

work page doi:10.4213/tm4284 2022
[13]

Porter, G. J. , TITLE =. Amer. J. Math. , VOLUME =. 1966 , PAGES =. doi:10.2307/2373148 , URL =

work page doi:10.2307/2373148 1966
[14]

and Suciu, A

Denham, G. and Suciu, A. I. , year=. Moment-angle Complexes, Monomial Ideals and Massey Products , volume=. Pure Appl. Math. Q. , publisher=. doi:10.4310/pamq.2007.v3.n1.a2 , number=

work page doi:10.4310/pamq.2007.v3.n1.a2 2007
[15]

, TITLE =

Amelotte, S. , TITLE =. Toric topology and polyhedral products , SERIES =. [2024] 2024 , ISBN =. doi:10.1007/978-3-031-57204-3\_1 , URL =

work page doi:10.1007/978-3-031-57204-3 2024
[16]

Theriault , booktitle =

S. Theriault , booktitle =. Toric homotopy theory , publisher =. 2017 , pages =

work page 2017
[17]

Ayzenberg, A. A. , TITLE =. Trans. Moscow Math. Soc. , YEAR =. doi:10.1090/s0077-1554-2014-00224-7 , URL =

work page doi:10.1090/s0077-1554-2014-00224-7 2014
[19]

and Bendersky, M

Bahri, A. and Bendersky, M. and Cohen, F. R. and Gitler, S. , TITLE =. Bol. Soc. Mat. Mex. (3) , VOLUME =. 2017 , NUMBER =. doi:10.1007/s40590-016-0124-8 , URL =

work page doi:10.1007/s40590-016-0124-8 2017
[20]

and Bendersky, M

Bahri, A. and Bendersky, M. and Cohen, F. R. and Gitler, S. , TITLE =. Adv. Math. , VOLUME =. 2010 , NUMBER =. doi:10.1016/j.aim.2010.03.026 , URL =

work page doi:10.1016/j.aim.2010.03.026 2010
[21]

Staniforth , publisher =

M. Staniforth , publisher =. The homotopy theory of polyhedral products , school =. 2023 , url =

work page 2023
[22]

Iriye and D

K. Iriye and D. Kishimoto , keywords =. Decompositions of polyhedral products for shifted complexes , journal =. 2013 , issn =. doi:https://doi.org/10.1016/j.aim.2013.05.003 , url =

work page doi:10.1016/j.aim.2013.05.003 2013
[23]

2015 , eprint=

Moment-angle manifolds and Panov's problem , author=. 2015 , eprint=

work page 2015
[24]

2024 , note=

Loop homology of moment-angle complexes in the flag case , author=. 2024 , note=

work page 2024
[25]

Discrete Comput

Spherical complexes and nonprojective toric varieties , volume=. Discrete Comput. Geom. , author=. 1986 , month=. doi:10.1007/bf02187689 , number=

work page doi:10.1007/bf02187689 1986
[26]

Homology, Homotopy and Applications , VOLUME =

Operations on polyhedral products and a new topological construction of infinite families of toric manifolds , author=. Homology, Homotopy and Applications , VOLUME =. 2015 , NUMBER =. doi:10.4310/HHA.2015.v17.n2.a8 , URL =

work page doi:10.4310/hha.2015.v17.n2.a8 2015
[27]

Proceedings of the Edinburgh Mathematical Society , author=

Loop space decompositions of moment-angle complexes associated to two-dimensional simplicial complexes , volume=. Proceedings of the Edinburgh Mathematical Society , author=. 2025 , pages=. doi:10.1017/S0013091525000203 , number=

work page doi:10.1017/s0013091525000203 2025
[28]

Buchstaber and T

V. Buchstaber and T. Panov , TITLE =. 2015 , PAGES =. doi:10.1090/surv/204 , URL =

work page doi:10.1090/surv/204 2015
[29]

and Panov, T

Grbi\'c, J. and Panov, T. and Theriault, S. and Wu, J. , TITLE =. Trans. Amer. Math. Soc. , VOLUME =. 2016 , NUMBER =. doi:10.1090/tran/6578 , URL =

work page doi:10.1090/tran/6578 2016
[30]

, TITLE =

Vidaurre, E. , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2018 , NUMBER =. doi:10.4310/HHA.2018.v20.n2.a13 , URL =

work page doi:10.4310/hha.2018.v20.n2.a13 2018
[31]

2020 , note=

On algebraic and combinatorial properties of weighted simplicial complexes , author=. 2020 , note=

work page 2020
[32]

2024 , note=

A stability theorem for bigraded persistence barcodes , author=. 2024 , note=

work page 2024
[33]

and Davis, M

Avramidi, G. and Davis, M. W. and Okun, B. and Schreve, K. , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2016 , NUMBER =. doi:10.1112/blms/bdv083 , URL =

work page doi:10.1112/blms/bdv083 2016
[34]

2024 , eprint=

Bier spheres and toric topology , author=. 2024 , eprint=

work page 2024
[35]

2024 , note =

Steenrod operations on polyhedral products , author=. 2024 , note =

work page 2024
[36]

and Okun, B

Davis, M. and Okun, B. , TITLE =. Groups Geom. Dyn. , VOLUME =. 2012 , NUMBER =. doi:10.4171/GGD/164 , URL =

work page doi:10.4171/ggd/164 2012
[37]

2023 , eprint=

On the Connectivity of the Vietoris-Rips Complex of a Hypercube Graph , author=. 2023 , eprint=

work page 2023
[38]

Loop spaces of polyhedral products associated with the polyhedral join product , volume=. Proc. Edinb. Math. Soc. (2) , author=. 2026 , pages=. doi:10.1017/S0013091525101223 , number=

work page doi:10.1017/s0013091525101223 2026
[39]

Provan, J. S. and Billera, L. J. , TITLE =. Math. Oper. Res. , FJOURNAL =. 1980 , NUMBER =. doi:10.1287/moor.5.4.576 , URL =

work page doi:10.1287/moor.5.4.576 1980
[40]

V. M. Buchstaber and T. E. Panov , year=. Torus actions, equivariant moment-angle complexes, and coordinate subspace arrangements , journal =. math/9912199 , archivePrefix=

work page arXiv
[41]

2024 , eprint=

Polyhedral products associated to pseudomanifolds , author=. 2024 , eprint=

work page 2024
[42]

2017 , eprint=

Complement Spaces, Dual Complexes and Polyhedral Product Spaces , author=. 2017 , eprint=

work page 2017
[43]

and Lutz, F

Joswig, M. and Lutz, F. H. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.jcta.2004.09.009 , URL =

work page doi:10.1016/j.jcta.2004.09.009 2005
[44]

Stanton, L. , year=. Loop space decompositions of highly symmetric spaces with applications to polyhedral products , volume=. European Journal of Mathematics , publisher=. doi:10.1007/s40879-023-00701-5 , number=

work page doi:10.1007/s40879-023-00701-5
[45]

Félix and D

Y. Félix and D. Tanré , journal =. Rational Homotopy of the Polyhedral Product Functor , urldate =

work page
[46]

and Bendersky, M

Bahri, A. and Bendersky, M. and Cohen, F. R. , TITLE =. Handbook of homotopy theory , SERIES =. [2020] 2020 , ISBN =

work page 2020
[47]

Grbic and S

J. Grbic and S. Theriault , title =. Russian Mathematical Surveys , pages =. 2016 , url =

work page 2016
[48]

Hilton, P. J. , title =. J. Lond. Math. Soc. , volume =. 1955 , month =. doi:10.1112/jlms/s1-30.2.154 , url =

work page doi:10.1112/jlms/s1-30.2.154 1955
[49]

, author=

The Construction FK. , author=. 1956 , publisher=

work page 1956
[50]

Anick, D. J. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1992 , NUMBER =. doi:10.2307/2154489 , URL =

work page doi:10.2307/2154489 1992
[51]

C. A. McGibbon and C. W. Wilkerson , journal =. Loop Spaces of Finite Complexes at Large Primes , urldate =

work page
[52]

Anick, D. J. , TITLE =. Algebraic topology (. 1989 , ISBN =. doi:10.1007/BFb0085216 , URL =

work page doi:10.1007/bfb0085216 1989
[53]

Cohen, F. R. and Moore, J. C. and Neisendorfer, J. A. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/1971269 , URL =

work page doi:10.2307/1971269 1979
[54]

Cohen, F. R. and Moore, J. C. and Neisendorfer, J. A. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/1971238 , URL =

work page doi:10.2307/1971238 1979
[55]

J. A. Neisendorfer , publisher =. II. The Exponent of a Moore Space , booktitle =. 1988 , lastchecked =. doi:doi:10.1515/9781400882113-003 , isbn =

work page doi:10.1515/9781400882113-003 1988
[56]

, TITLE =

Serre, J.P. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1951 , PAGES =. doi:10.2307/1969485 , URL =

work page doi:10.2307/1969485 1951
[57]

1965 , issn =

Higher order Whitehead products , journal =. 1965 , issn =. doi:https://doi.org/10.1016/0040-9383(65)90039-X , url =

work page doi:10.1016/0040-9383(65)90039-x 1965
[58]

G. J. H. Simmons , publisher =. Homotopy theory of polyhedral products , school =. 2023 , url =

work page 2023
[59]

and Kishimoto, D

Iriye, K. and Kishimoto, D. , year=. Fat-wedge filtration and decomposition of polyhedral products , volume=. Kyoto Journal of Mathematics , publisher=. doi:10.1215/21562261-2017-0038 , number=

work page doi:10.1215/21562261-2017-0038 2017
[60]

Theriault , title =

S. Theriault , title =. Topology, Geometry, Combinatorics, and Mathematical Physics , note =. 2024 , pages =

work page 2024
[61]

and Theriault, S

Huang, R. and Theriault, S. , title =. Ann. Inst. Fourier (Grenoble) , volume =. 2024 , pages =

work page 2024
[62]

Boyde, Guy , title =. Math. Z. , volume =. 2022 , doi =

work page 2022
[63]

Neisendorfer, J. A. and Selick, P. , title =. Current Trends in Algebraic Topology, Part 1 , address =. 1981 , pages =

work page 1981
[64]

Kim, J. H. , title =. Bull. Korean Math. Soc. , volume =. 2018 , doi =

work page 2018
[65]

2026 , eprint=

Anick's conjecture for polyhedral products , author=. 2026 , eprint=

work page 2026
[66]

On conjectures of Moore and Serre in the case of torsion-free suspensions , volume=. Math. Proc. Cambridge Philos. Soc. , author=. 1983 , pages=. doi:10.1017/S0305004100060916 , number=

work page doi:10.1017/s0305004100060916 1983
[67]

Moore’s conjecture for connected sums , volume=. Canad. Math. Bull. , author=. 2024 , pages=. doi:10.4153/S0008439523000930 , number=

work page doi:10.4153/s0008439523000930 2024
[68]

and Halperin, S

Félix, Y. and Halperin, S. and Thomas, J.C. , title =. 2001 , doi =

work page 2001
[69]

I. M. James , journal =. The Suspension Triad of a Sphere , urldate =

work page
[70]

, title =

Toda, H. , title =. J. Inst. Polytech. Osaka City Univ. Ser. A , volume =. 1956 , pages =

work page 1956
[71]

, title =

Long, J. , title =. 1978 , type =

work page 1978
[72]

and Pitsch, W

Chachólski, W. and Pitsch, W. and Scherer, J. and Stanley, D. , title =. Int. Math. Res. Not. IMRN , volume =. 2008 , pages =

work page 2008
[73]

, title =

Stelzer, M. , title =. Topology , volume =. 2004 , pages =

work page 2004
[74]

Anick, D. J. , title =. Algebraic Topology (Arcata, CA, 1986) , series =

work page 1986
[75]

1989 , issn =

The image of the stable J-homomorphism , journal =. 1989 , issn =. doi:https://doi.org/10.1016/0040-9383(89)90031-1 , url =

work page doi:10.1016/0040-9383(89)90031-1 1989
[76]

Eldridge , publisher =

B. Eldridge , publisher =. Polyhedral products associated with polyhedral join products , school =. 2025 , url =

work page 2025