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arxiv: 2412.08260 · v2 · submitted 2024-12-11 · 🧮 math.AG · math.GR· math.GT

Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Francesco Polizzi , Pietro Sabatino This is my paper

Pith reviewed 2026-05-23 07:38 UTC · model grok-4.3

classification 🧮 math.AG math.GRmath.GT
keywords pure braid groupsKodaira fibrationsfinite group quotientsfundamental groupsRiemann surfacessurface fibrationsalgebraic geometry
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The pith

Finite quotients of the pure braid group on two strands are at least order 64 except for order 32, with equality cases classified, yielding double Kodaira fibrations with same invariants but different fundamental groups.

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

The authors examine finite groups G that are quotients of the pure braid group P2 on a Riemann surface of genus b, but that do not arise from the fundamental group of the product of two such surfaces. They use diagonal double Kodaira structures to prove that any such G either has order 32 or at least 64, and they list all groups achieving the bound of 64. This algebraic result is applied to construct two three-dimensional families of double Kodaira fibrations that agree in their biregular invariants and Betti numbers yet have non-isomorphic fundamental groups.

Core claim

If G is a finite quotient of P2(Σb) not factoring through π1(Σb × Σb) and admitting a diagonal double Kodaira structure, then |G| is either 32 or at least 64; the groups of order 64 are completely classified. This classification produces two 3-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental groups.

What carries the argument

Diagonal double Kodaira structures, which are special systems of generators on finite groups that correspond to quotients of the pure braid group P2(Σb).

If this is right

  • Such groups have no possible orders between 33 and 63.
  • All groups of order 64 with diagonal double Kodaira structures are listed explicitly.
  • Double Kodaira fibrations can be constructed in 3-dimensional families that are indistinguishable by biregular invariants or Betti numbers.
  • The fundamental group distinguishes these fibrations where numerical invariants do not.

Where Pith is reading between the lines

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

  • The result indicates that the fundamental group captures more information than the usual numerical invariants for these fibrations.
  • Similar order bounds might apply to other braid group quotients or other types of surface fibrations.
  • Explicit lists of groups of order 64 could be used to enumerate possible fundamental groups in this geometric setting.

Load-bearing premise

The groups in question are required to admit diagonal double Kodaira structures as defined in prior work.

What would settle it

Finding a group of order between 33 and 63 that admits a diagonal double Kodaira structure and is a quotient of P2(Σb) not coming from the product surface would disprove the order bound.

read the original abstract

Let $\Sigma_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$ which do not factor through $\pi_1(\Sigma_b \times \Sigma_b)$. Building on our previous work on some special systems of generators on finite groups that we called \emph{diagonal double Kodaira structures}, we prove that, if $G$ has not order $32$, then $|G| \geq 64$, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group.

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 / 0 minor

Summary. The paper investigates finite quotients G of the pure braid group P_2(Σ_b) that do not factor through π_1(Σ_b × Σ_b). Using the notion of diagonal double Kodaira structures from prior work, it proves that such G have order either 32 or at least 64, gives a complete classification of all cases with |G|=64, and constructs two explicit 3-dimensional families of double Kodaira fibrations that share the same biregular invariants and Betti numbers but possess non-isomorphic fundamental groups.

Significance. If the algebraic classification and geometric constructions hold, the work supplies a sharp order bound together with an exhaustive list at |G|=64 for these braid-group quotients, which directly constrains the possible fundamental groups of double Kodaira fibrations. The two families furnish concrete examples where standard invariants fail to distinguish homeomorphism types, contributing concrete data to questions of rigidity and moduli in the theory of surface fibrations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation proceeds via direct algebraic analysis of quotients of the pure braid group P2(Σb) that admit diagonal double Kodaira structures (defined in prior work) and do not factor through π1(Σb × Σb). The order bound |G| ≥ 64 (when not 32) and the exhaustive classification at |G|=64 are established in this manuscript through case analysis of the braid-group quotients; the geometric constructions of the two 3-dimensional families are likewise explicit and independent. The reference to prior work supplies only the definition of the structures and does not reduce any stated prediction or bound to a fitted input or self-referential equation by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5677 in / 1252 out tokens · 24071 ms · 2026-05-23T07:38:52.470848+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    [At69] M. F. Atiyah: The signature of fibre bundles, in Global Analysis (Papers in honor of K. Kodaira) 73–84, Univ. Tokyo Press (1969). (Cited on p

    work page 1969

  2. [2]

    Bryan, R

    [BD02] J. Bryan, R. Donagi: Surface bundles over surfaces of small genus, Geom. Topol. 6 (2002), 59–67. (Cited on p

    work page 2002

  3. [3]

    Bryan, R

    24 FRANCESCO POLIZZI AND PIETRO SABATINO [BDS01] J. Bryan, R. Donagi, A. I. Stipsicz: Surface bundles: some interesting examples,Turkish J. Math.25 (2001), no. 1, 61–68. (Cited on p

    work page 2001

  4. [4]

    Catanese: Kodaira fibrations and beyond: methods for moduli theory, Japan

    [Cat17] F. Catanese: Kodaira fibrations and beyond: methods for moduli theory, Japan. J. Math. 12 (2017), no. 2, 91–174. (Cited on p

    work page 2017

  5. [5]

    Catanese, S

    [CatRol09] F. Catanese, S. Rollenske: Double Kodaira fibrations,J. Reine Angew. Math.628 (2009), 205–233. (Cited on p

    work page 2009

  6. [6]

    Causin, F

    [CaPol21] A. Causin, F. Polizzi: Surface braid groups, finite Heisenberg covers and double Kodaira fibrations, Ann. Sc. Norm. Super. Pisa Cl. Sci.Vol. XXII (2021), 1309-1352 (Cited on p. 1, 16,

    work page 2021

  7. [7]

    Endo: A construction of surface bundles over surfaces with non-zero signature, Osaka J

    [En98] H. Endo: A construction of surface bundles over surfaces with non-zero signature, Osaka J. Math.35 (1998), 915–930. (Cited on p

    work page 1998

  8. [8]

    [EKKOS02] H. Endo, M. Korkmaz, D, Kotschick, B. Ozbagci, A. Stipsicz: Commutators, Lefschetz fibrations and the signature of surface bundles, Topology 41 no. 5 (2002), 961–977. (Cited on p

    work page 2002

  9. [9]

    [Fox57] R. H. Fox: Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press (1957), 243–257 (Cited on p

    work page 1957

  10. [10]

    [GG04] D. L. Gonçalves, J. Guaschi: On the structure of surface pure braid groups,J. Pure Appl. Algebra186 (2004), 187–218. (Cited on p. 2,

    work page 2004

  11. [11]

    Hirzebruch: The signature of ramified covers, inGlobal Analysis (Papers in honor of K

    [Hir69] F. Hirzebruch: The signature of ramified covers, inGlobal Analysis (Papers in honor of K. Kodaira) 253–265, Univ. Tokyo Press (1969). (Cited on p

    work page 1969

  12. [12]

    Kas: On deformations of a certain type of irregular algebraic surface, Amer

    [Kas68] A. Kas: On deformations of a certain type of irregular algebraic surface, Amer. J. Math.90 (1968), 789–804. (Cited on p

    work page 1968

  13. [13]

    LeBrun: Diffeomorphisms, symplectic forms and Kodaira fibrations, Geom

    [LeBrun00] C. LeBrun: Diffeomorphisms, symplectic forms and Kodaira fibrations, Geom. Topol. 4 (2000), 451–456. (Cited on p

    work page 2000

  14. [14]

    A Lee: Surface bundles over surfaces with a fixed signature, J

    [L17] J. A Lee: Surface bundles over surfaces with a fixed signature, J. Korean Math. Soc.54 (2017), no. 2, 545–561. (Cited on p

    work page 2017

  15. [15]

    [LLR20] J. A. Lee, M. Lönne, S. Rollenske: Double Kodaira fibrations with small signature, Internat. J. Math. 31 (2020), no. 7, 2050052, 42 pp. (Cited on p

    work page 2020

  16. [16]

    Polizzi: Diagonal double Kodaira structures on finite groups, Current Trends in Analysis, its Applications and Computation, Trends in Mathematics, Birkäuser, Cham (2022), 111-128

    [Pol20] F. Polizzi: Diagonal double Kodaira structures on finite groups, Current Trends in Analysis, its Applications and Computation, Trends in Mathematics, Birkäuser, Cham (2022), 111-128. (Cited on p

    work page 2022

  17. [17]

    Dedicata 216, 65 (2022) (Cited on p

    [PolSab22] Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature, Geom. Dedicata 216, 65 (2022) (Cited on p. 1, 2, 5, 11, 12, 14, 15, 16,

    work page 2022

  18. [18]

    Rollenske: Compact moduli for certain Kodaira fibrations, Ann

    [Rol10] S. Rollenske: Compact moduli for certain Kodaira fibrations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 851-874. (Cited on p

    work page 2010

  19. [19]

    [St02] A. I. Stipsicz: Surface bundles with nonvanishing signature, Acta Math. Hungar. 95 (2002), no. 4, 299–307. (Cited on p

    work page 2002

  20. [20]

    Zaal: Explicit complete curves in the moduli space of curves of genus three, Geom

    [Zaal95] C. Zaal: Explicit complete curves in the moduli space of curves of genus three, Geom. Dedicata 56 (1995), 185–196 (Cited on p

    work page 1995