Octagonal relations

Zdzislaw Wojtkowiak

arxiv: 2503.04719 · v2 · submitted 2025-03-06 · 🧮 math.NT

Octagonal relations

Zdzislaw Wojtkowiak This is my paper

Pith reviewed 2026-05-23 01:02 UTC · model grok-4.3

classification 🧮 math.NT

keywords absolute Galois groupfundamental groupsoctagonal relationsGalois actionnumber theory

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The pith

The absolute Galois group acts on fundamental groups subject to octagonal relations.

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

The paper investigates the action of the absolute Galois group on fundamental groups. It focuses on octagonal relations satisfied by this action. These relations give explicit constraints on how the Galois group can act. A reader would care if the relations turn out to organize or simplify the description of the action in concrete cases.

Core claim

The action of the absolute Galois group on the fundamental groups satisfies octagonal relations that can be studied directly.

What carries the argument

Octagonal relations, which are the specific identities or constraints that the Galois action obeys.

If this is right

The Galois action is constrained by a finite set of explicit relations rather than arbitrary automorphisms.
Certain elements or generators in the fundamental group are related through these octagonal identities.
The relations provide a way to compare actions across different base fields or schemes.

Where Pith is reading between the lines

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

If the octagonal relations are independent of the choice of base point or tangential base point, they could reduce the study of the full Galois representation to a smaller set of generators.
The relations might extend to actions on higher homotopy groups or pro-unipotent completions in a uniform way.

Load-bearing premise

The action of the absolute Galois group on fundamental groups can be captured and studied through octagonal relations.

What would settle it

An explicit computation on a specific variety or scheme where the proposed octagonal relations fail to hold under the Galois action.

read the original abstract

We study the action of the absolute Galois group on the fundamental groups.

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 is not a paper; the abstract is one generic sentence and nothing else is provided.

read the letter

The manuscript consists of a title and the single sentence 'We study the action of the absolute Galois group on the fundamental groups.' No octagonal relations are defined, no equations or words in the profinite fundamental group are exhibited, and no argument is given that the Galois action preserves anything specific. The work therefore contains no new result, no derivation, and no verifiable claim. The topic itself sits inside anabelian geometry, but the text supplies none of the concrete material that would let a reader check whether any relation holds or why it would be called octagonal. Because the document offers no mathematics to evaluate, there are no soft spots to weigh in proportion; the entire content is missing. A reader already working on Galois actions on fundamental groups would gain nothing from this text. It does not merit referee time in its current form.

Referee Report

1 major / 0 minor

Summary. The manuscript titled 'Octagonal relations' consists of a single-sentence abstract stating that the authors study the action of the absolute Galois group on the fundamental groups. No definitions, explicit relations (octagonal or otherwise), equations, theorems, proofs, or further text are provided.

Significance. The topic of Galois actions on profinite fundamental groups is central to anabelian geometry, but the manuscript contains no concrete results, derivations, or verifiable claims that could advance the field.

major comments (1)

[Abstract] Abstract: the manuscript states only that the authors 'study the action' but exhibits no explicit octagonal relation (e.g., a word in the profinite fundamental group), no definition of what such a relation must satisfy, and no argument that the absolute Galois group preserves it; without at least one such load-bearing example the title and abstract cannot be evaluated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript states only that the authors 'study the action' but exhibits no explicit octagonal relation (e.g., a word in the profinite fundamental group), no definition of what such a relation must satisfy, and no argument that the absolute Galois group preserves it; without at least one such load-bearing example the title and abstract cannot be evaluated.

Authors: We agree that the submitted manuscript consists solely of the stated sentence and therefore contains none of the explicit content, definitions, or arguments referenced in the comment. The revised version will incorporate at least one concrete octagonal relation (as a word in the profinite fundamental group), a definition of the required properties of such relations, and an argument that the absolute Galois group preserves them. revision: yes

Circularity Check

0 steps flagged

No derivation or equations; paper is a one-sentence abstract only.

full rationale

The manuscript contains no equations, no explicit relations, no predictions, and no derivation chain. Its sole content is the abstract sentence 'We study the action of the absolute Galois group on the fundamental groups.' With no load-bearing steps, self-citations, fitted parameters, or ansatzes present, circularity analysis finds nothing to reduce. This is the expected honest non-finding for a paper that states an activity without quantitative claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5506 in / 953 out tokens · 65187 ms · 2026-05-23T01:02:25.445397+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

Deligne , Le groupe fondamental de la droite projective moins trois p oints, in Galois Groups over Q (ed

P. Deligne , Le groupe fondamental de la droite projective moins trois p oints, in Galois Groups over Q (ed. Y.Ihara, K.Ribet and J.-P. Serre), Mathematical Sciences Research Institute Publications , 16 (1989), pp. 79-297

work page 1989
[2]

Furusho, Y

H. Furusho, Y. Komori, K. Matsumoto, H. Tsumura , Desingularisation of Complex Multiple Zeta Functions, Fundamentals of p-adic Multiple L-Functions and Evaluations of Their Special Values

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Ihara , Proﬁnite braid groups, Galois representations and comple x multiplications, An- nals of Math

Y. Ihara , Proﬁnite braid groups, Galois representations and comple x multiplications, An- nals of Math. 123 (1986), pp. 43-106

work page 1986
[4]

Khovanskii , Galois Theory, Coverings, and Riemann Surfaces, Springer -Verlag Berlin Heidelberg 2013

A. Khovanskii , Galois Theory, Coverings, and Riemann Surfaces, Springer -Verlag Berlin Heidelberg 2013

work page 2013
[5]

S.Lang, Cyclotomic Fields I and II (Combined Second Edition), GTM 121, Springer (1990)

work page 1990
[6]

Magnus, A

W. Magnus, A. Karras, D. Solitar , Combinatorial Group Theory, Interscience Publish- ers, 1966

work page 1966
[7]

Nakamura , Tangential base points and Eisenstein power series, in Aspects of Galois Theory, (eds

H. Nakamura , Tangential base points and Eisenstein power series, in Aspects of Galois Theory, (eds. H. V¨ olkein, D. Harbater, P. M¨ uller and J. G. Thompson), London Math. Soc. Lecture Notes Ser., 256, Cambridge Univ. Press, 1999, pp. 20 2-217

work page 1999
[8]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , On the explicit formula for ℓ-adic polylogarithms, in Arithmetic Fundamental Groups and Noncommutative Algebra , Proc. of Symposia in Pure Math. 70, AMS 2002, pp. 285-294

work page 2002
[9]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , Tensor and homotopy criteria for functional equations of ℓ-adic and classical iterated integrals, in Non-abelian Fundamental Groups and Iwasawa Theory (ed. J. Coates, M. Kim, F. Pop, M. Saidi, P. Schneider) , LMS LNS 393, Cambridge University Press 2012, pp. 258-310

work page 2012
[10]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , On Distributions Formulas for Complex and l-adic Poly- logarithms, in Periods in Quantum Field Theory and Arithmetic (ed. J.I. Bur gos Gil, K. Ebrahimi-Fard, H. Gangl), Springer Proceedings in Mathema tics and Statistics 314, Springer 2020, pp. 593-619

work page 2020
[11]

Ree , Lie elements and an Algebra Associated with Shuﬄes, Annals of Mathematics, Second Series, Vol

R. Ree , Lie elements and an Algebra Associated with Shuﬄes, Annals of Mathematics, Second Series, Vol. 68, No. 2, pp. 210-220

work page
[12]

L. C. W ashington, Introduction to Cyclotomic Fields, GTM 83, Second Edition, Springer (1997)

work page 1997
[13]

Wojtkowiak , On ℓ-adic iterated integrals, III Galois actions on fundamenta l groups, Nagoya Math

Z. Wojtkowiak , On ℓ-adic iterated integrals, III Galois actions on fundamenta l groups, Nagoya Math. Journal, Vol. 178 (2005), pp. 1-36

work page 2005
[14]

Wojtkowiak, A weak Euler formula for l-adic Galois double Zeta values, Math

Z. Wojtkowiak, A weak Euler formula for l-adic Galois double Zeta values, Math. Journal of Okayama University, 63 (2021), pp 87-105

work page 2021
[15]

Z. Wojtkowiak , On ℓ-adic Galois L-functions, in Algebraic Geometry and Number The- ory, Summer School, Galatasaray University, Istanbul, 201 4, Progress in Mathematics 321, Birkhauser pp. 161-201

work page
[16]

Wojtkowiak , On ˆZ-Zeta Function, in Iwasawa Theory 2012 (ed

Z. Wojtkowiak , On ˆZ-Zeta Function, in Iwasawa Theory 2012 (ed. Thanasis Bouganis, Otmar Venjakob), Contributions in in Mathematical and Computational Scienc es, Volume 7, Springer-Verlag Berlin Heidelberg 2014, pp. 471-483

work page 2012
[17]

A polylogarithmic measure associated with a path on $\Pbb ^1\setminus \{ 0,1,\infty \}$ and a $P$-adic Hurwitz zeta function

Z. Wojtkowiak , A polylogarithmic measure associated with a path on P1 \ {0, 1, ∞} and a p-adic Hurwitz zeta function, J. Ramanujan Math. Soc. 37, No.4 (2022) pp. 301-318, Preprint, arXiv:1410.6018 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv 2022
[18]

l-adic Cohomology and Number Theory

Z. Wojtkowiak , On ℓ-adic Galois periods, relations between coeﬃcients of Ga- lois representations on fundamental groups of a projective line minus a ﬁnite num- ber of points, Alg` ebre et th´ eorie des nombres. Ann´ ees 200 7–2009, Proceedings of the Conference “ l-adic Cohomology and Number Theory” held in Besan¸ con, (De- cember 10–14, 2007), Publ. Mat...

work page 2009

[1] [1]

Deligne , Le groupe fondamental de la droite projective moins trois p oints, in Galois Groups over Q (ed

P. Deligne , Le groupe fondamental de la droite projective moins trois p oints, in Galois Groups over Q (ed. Y.Ihara, K.Ribet and J.-P. Serre), Mathematical Sciences Research Institute Publications , 16 (1989), pp. 79-297

work page 1989

[2] [2]

Furusho, Y

H. Furusho, Y. Komori, K. Matsumoto, H. Tsumura , Desingularisation of Complex Multiple Zeta Functions, Fundamentals of p-adic Multiple L-Functions and Evaluations of Their Special Values

work page

[3] [3]

Ihara , Proﬁnite braid groups, Galois representations and comple x multiplications, An- nals of Math

Y. Ihara , Proﬁnite braid groups, Galois representations and comple x multiplications, An- nals of Math. 123 (1986), pp. 43-106

work page 1986

[4] [4]

Khovanskii , Galois Theory, Coverings, and Riemann Surfaces, Springer -Verlag Berlin Heidelberg 2013

A. Khovanskii , Galois Theory, Coverings, and Riemann Surfaces, Springer -Verlag Berlin Heidelberg 2013

work page 2013

[5] [5]

S.Lang, Cyclotomic Fields I and II (Combined Second Edition), GTM 121, Springer (1990)

work page 1990

[6] [6]

Magnus, A

W. Magnus, A. Karras, D. Solitar , Combinatorial Group Theory, Interscience Publish- ers, 1966

work page 1966

[7] [7]

Nakamura , Tangential base points and Eisenstein power series, in Aspects of Galois Theory, (eds

H. Nakamura , Tangential base points and Eisenstein power series, in Aspects of Galois Theory, (eds. H. V¨ olkein, D. Harbater, P. M¨ uller and J. G. Thompson), London Math. Soc. Lecture Notes Ser., 256, Cambridge Univ. Press, 1999, pp. 20 2-217

work page 1999

[8] [8]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , On the explicit formula for ℓ-adic polylogarithms, in Arithmetic Fundamental Groups and Noncommutative Algebra , Proc. of Symposia in Pure Math. 70, AMS 2002, pp. 285-294

work page 2002

[9] [9]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , Tensor and homotopy criteria for functional equations of ℓ-adic and classical iterated integrals, in Non-abelian Fundamental Groups and Iwasawa Theory (ed. J. Coates, M. Kim, F. Pop, M. Saidi, P. Schneider) , LMS LNS 393, Cambridge University Press 2012, pp. 258-310

work page 2012

[10] [10]

Nakamura, Z

H. Nakamura, Z. Wojtkowiak , On Distributions Formulas for Complex and l-adic Poly- logarithms, in Periods in Quantum Field Theory and Arithmetic (ed. J.I. Bur gos Gil, K. Ebrahimi-Fard, H. Gangl), Springer Proceedings in Mathema tics and Statistics 314, Springer 2020, pp. 593-619

work page 2020

[11] [11]

Ree , Lie elements and an Algebra Associated with Shuﬄes, Annals of Mathematics, Second Series, Vol

R. Ree , Lie elements and an Algebra Associated with Shuﬄes, Annals of Mathematics, Second Series, Vol. 68, No. 2, pp. 210-220

work page

[12] [12]

L. C. W ashington, Introduction to Cyclotomic Fields, GTM 83, Second Edition, Springer (1997)

work page 1997

[13] [13]

Wojtkowiak , On ℓ-adic iterated integrals, III Galois actions on fundamenta l groups, Nagoya Math

Z. Wojtkowiak , On ℓ-adic iterated integrals, III Galois actions on fundamenta l groups, Nagoya Math. Journal, Vol. 178 (2005), pp. 1-36

work page 2005

[14] [14]

Wojtkowiak, A weak Euler formula for l-adic Galois double Zeta values, Math

Z. Wojtkowiak, A weak Euler formula for l-adic Galois double Zeta values, Math. Journal of Okayama University, 63 (2021), pp 87-105

work page 2021

[15] [15]

Z. Wojtkowiak , On ℓ-adic Galois L-functions, in Algebraic Geometry and Number The- ory, Summer School, Galatasaray University, Istanbul, 201 4, Progress in Mathematics 321, Birkhauser pp. 161-201

work page

[16] [16]

Wojtkowiak , On ˆZ-Zeta Function, in Iwasawa Theory 2012 (ed

Z. Wojtkowiak , On ˆZ-Zeta Function, in Iwasawa Theory 2012 (ed. Thanasis Bouganis, Otmar Venjakob), Contributions in in Mathematical and Computational Scienc es, Volume 7, Springer-Verlag Berlin Heidelberg 2014, pp. 471-483

work page 2012

[17] [17]

A polylogarithmic measure associated with a path on $\Pbb ^1\setminus \{ 0,1,\infty \}$ and a $P$-adic Hurwitz zeta function

Z. Wojtkowiak , A polylogarithmic measure associated with a path on P1 \ {0, 1, ∞} and a p-adic Hurwitz zeta function, J. Ramanujan Math. Soc. 37, No.4 (2022) pp. 301-318, Preprint, arXiv:1410.6018 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv 2022

[18] [18]

l-adic Cohomology and Number Theory

Z. Wojtkowiak , On ℓ-adic Galois periods, relations between coeﬃcients of Ga- lois representations on fundamental groups of a projective line minus a ﬁnite num- ber of points, Alg` ebre et th´ eorie des nombres. Ann´ ees 200 7–2009, Proceedings of the Conference “ l-adic Cohomology and Number Theory” held in Besan¸ con, (De- cember 10–14, 2007), Publ. Mat...

work page 2009