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arxiv: 2606.24378 · v1 · pith:7C3SQTXNnew · submitted 2026-06-23 · 🧮 math.QA · math.GT

Central extensions of mapping class groups of surfaces from stated skein algebras

Joris Moulai This is my paper

Pith reviewed 2026-06-25 21:50 UTC · model grok-4.3

classification 🧮 math.QA math.GT
keywords mapping class groupscentral extensionsstated skein algebrasribbon Hopf algebrasprojective representationssurfacesquantum topology
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The pith

The stated skein algebra of a surface paired with a factorizable ribbon Hopf algebra determines an explicit central extension of the mapping class group.

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

The paper computes the central extension of the mapping class group of a surface that is induced by the projective representation coming from the stated skein algebra of the surface together with a finite-dimensional factorizable ribbon Hopf algebra. The surfaces considered have genus g, zero or one boundary component, and n marked points. The proof proceeds entirely in two dimensions and avoids any appeal to three-dimensional topological quantum field theory. A reader would care because central extensions of mapping class groups control projective representations that arise throughout quantum topology and low-dimensional geometry.

Core claim

We compute the central extension of the mapping class group of Σ, associated to the projective representation defined from the stated skein algebra of Σ and H, for a surface Σ of genus g with zero or one boundary component and n marked points, and H a finite-dimensional factorizable ribbon Hopf algebra. Our proof is purely two-dimensional, and makes no use of TQFT arguments.

What carries the argument

The stated skein algebra of Σ with the Hopf algebra H, which supplies the projective representation whose associated central extension is computed.

If this is right

  • The central extension is determined directly from the two-dimensional skein data without three-dimensional input.
  • The result applies to every finite-dimensional factorizable ribbon Hopf algebra H.
  • The computation covers all surfaces of genus g with at most one boundary component and any number of marked points.
  • The extension class is independent of TQFT constructions.

Where Pith is reading between the lines

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

  • The same method could be checked on low-genus examples to produce concrete cocycle representatives.
  • Similar skein constructions for other algebraic structures might yield parallel central extensions.
  • The two-dimensional approach may simplify comparison with other known extensions arising from quantum invariants.

Load-bearing premise

The stated skein algebra of the surface with the given Hopf algebra produces a well-defined projective representation of the mapping class group.

What would settle it

An explicit computation of the central extension for the torus or the once-punctured sphere that differs from the class obtained from the stated skein algebra construction.

Figures

Figures reproduced from arXiv: 2606.24378 by Joris Moulai.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: Lantern relation The following relation is an immediate consequence of the lantern relation and the capping morphism ([FM12, Prop. 3.19]): Corollary 1.3 (Puncture relation). Let b1, b2, b3, x, y, and z be the simple closed curves arranged as in [PITH_FULL_IMAGE:figures/full_fig_p006_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: We have the following relation in Γ s g,n, called the puncture relation: τb1 τb2 τb3 = τxτyτz. We denote by rp := τb1 τb2 τb3 τ −1 z τ −1 y τ −1 x the associated relator. b3 b2 b1 x y z [PITH_FULL_IMAGE:figures/full_fig_p006_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: 3-chain relation We will use the following presentation of Γs g,n. Theorem 1.5 ( [Har83], [Ger96, Thm. B], [FK14, Lem. 2.7]). For all g ≥ 2, n ≥ 0, and s ≥ 0, the group Γ s g,n has the following presentation: • Generators: All Dehn twists τγ along non-separating simple closed curves γ ⊂ Σ s g,n. • Relations: (1) All 0 and 1-braid relations, (2) One lantern relation, if g ≥ 3, (3) One 3-chain relation, (4… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Standard curves in the surface Σ1 g,n bj aj mg+k [PITH_FULL_IMAGE:figures/full_fig_p011_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Standard curves in the ribbon surface presentation of Σ1 g,n These embeddings have the following properties: iaj , ibj , img+k (18) : L0,1(H) → Lg,n(H) are morphisms of algebras, ib1 (β1)ia1 (α1)· · · ibg (βg)iag (αg)img+1 (ψ1)· · · img+n (19) (ψn) = β1 ⊗ α1 ⊗ · · · ⊗ βg ⊗ αg ⊗ ψ1 ⊗ · · · ⊗ ψn, iai (φ)ibi (ψ) = X i,j,k,l ibi (r l rk ▷ ψ ◁ rirj )iai [PITH_FULL_IMAGE:figures/full_fig_p011_2_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Example of a simple closed based and positively oriented curve in the ribbon surface, with normalization N(z) = 3 − 3 = 0 An element γ ∈ π1(Σ1 g,n) is said to be simple if it is represented by a simple closed and based curve which we still denote by γ. Let γ ∈ π1(Σ1 g,n) be simple, see [PITH_FULL_IMAGE:figures/full_fig_p015_3_1.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Curves aj , bj , and cj in the surface Σ1 g,n. Proof. See [Fai26, Prop. 5.22] for the actions of ˆτaj and ˆτbj . The action of ˆτcj is computed in Appendix B.1 using the diagrammatic computation described in Appendix A.1. □ 3.3. The projective representation ρg,n of Γg,n. The goal of this section is to extend the results of Subsection 3.2 to surfaces without boundary component Σg,n. We will need the base… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Curve ∂g,n in the surface Σ1 g,n. The capping morphism ([FM12, Prop. 3.19]) and the Birman exact sequence ([FM12, Thm. 4.6]) imply that Γg,n is a quotient of Γ1 g,n. More precisely, we have the following composition of morphisms: (41) Γ1 g,n Cap −−→ Γg,n+1 Forget −−−−−→ Γg,n, where Push : π1(Σg,n+1) → Γg,n+1 is the map from the Birman exact sequence and: • ker(Cap) = ⟨τ∂g,n ⟩, • ker(Forget) = Push(π1(Σg,… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Curves of the star relation Therefore, we have: τ∂g,n ∈ ker ρg,n ⇐⇒ ρg,n (τα3 τα2 τα1 τβ) 3  = ρg,n(τδ1 τδ2 ). We will now prove the equality ρg,n (τα3 τα2 τα1 τβ) 3  = ρg,n(τδ1 τδ2 ). As based curves, we have α2− = α3+∂ −1 g,n, where α2− ∈ π1(Σ1 g,n) (resp. α3+ ∈ π1(Σ1 g,n)) denotes a representative of α2 (resp. α3) that is negatively (resp. positively) oriented (Prop. 3.4 implies that the orientation… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Example of curves whose union separates Σ1 g,n Lemma 4.3. Let (α1, β1) and (α2, β2) be two pairs of non-separating simple closed curves of Σ 1 g,n that do not intersect and are in the same position. Then we have: τˆα1 τˆβ1 τˆ −1 α1 τˆ −1 β1 = ˆτα2 τˆβ2 τˆ −1 α2 τˆ −1 β2 . Proof. Let (α1, β1) and (α2, β2) be two pairs of non-separating simple closed curves that do not intersect and are in the same positio… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p041_2.png] view at source ↗
read the original abstract

Let $\Sigma$ be a surface of genus $g$ with zero or one boundary component and $n$ marked points, and $H$ a finite-dimensional factorizable ribbon Hopf algebra. We compute the central extension of the mapping class group of $\Sigma$, associated to the projective representation defined from the stated skein algebra of $\Sigma$ and $H$. Our proof is purely two-dimensional, and makes no use of TQFT arguments.

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 computes the central extension of the mapping class group of a surface Σ (genus g, at most one boundary component, n marked points) associated to the projective representation arising from the stated skein algebra of Σ with a finite-dimensional factorizable ribbon Hopf algebra H. The argument is presented as purely two-dimensional and independent of TQFT constructions.

Significance. If correct, the result supplies an explicit 2D computation of the extension class in terms of stated skein data, which is of interest for relating quantum algebra constructions to mapping class group representations. The restriction to surfaces with ≤1 boundary component together with the factorizability and ribbon hypotheses on H is used to guarantee the projective action exists, and the avoidance of TQFT is a methodological strength.

minor comments (1)
  1. The abstract asserts the computation but contains no equations, lemmas, or outline of the extension class; adding a single sentence summarizing the form of the computed cocycle or the key 2D relation used would improve accessibility without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including recognition of the explicit 2D computation and the methodological choice to avoid TQFT. The report recommends minor revision but lists no major comments, so we have no specific points requiring rebuttal or clarification.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central computation of the central extension class proceeds from the definition of the stated skein algebra of Σ with factorizable ribbon Hopf algebra H, using only 2D arguments on surfaces with at most one boundary component. No derivation step reduces by construction to a fitted parameter, self-citation load-bearing premise, or renamed input; the projective representation is taken as given from the algebra's standard properties, and the extension is extracted directly without circular redefinition or imported uniqueness theorems from the same authors. The proof is explicitly self-contained against external 3D TQFT benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result is stated to rest on the standard properties of factorizable ribbon Hopf algebras and stated skein algebras.

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Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    On the quantum Teichm¨ uller invariants of fibred cusped 3- manifolds

    [BB18] S. Baseilhac and R. Benedetti. “On the quantum Teichm¨ uller invariants of fibred cusped 3- manifolds”.Geom. Dedicata197.1 (2018), 1–32. [BFR25] S. Baseilhac, M. Faitg, and P. Roche. “Noetherian and affine properties of quantum moduli and g-skein algebras”.Quantum Topol.(2025). [BFR26] S. Baseilhac, M. Faitg, and P. Roche.On the structure and repre...

    2018

  2. [2]

    On the structure and representations of quantum graph algebras at roots of unity

    arXiv:2601.08789. [BL07] F. Bonahon and X. Liu. “Representations of the quantum Teichm¨ uller space and invariants of surface diffeomorphisms”.Geom. Topol.11.2 (2007), 889–937. [Bla+95] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel. “Topological quantum field theories derived from the Kauffman bracket”.Topology34.4 (1995), 883–927. [Bro82] K. S. Brow...

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  3. [3]

    [CP00] V

    arXiv:2112.12852. [CP00] V. Chari and A. Pressley.A guide to quantum groups. Reprint. Cambridge University Press,

    work page arXiv

  4. [4]

    3-Dimensional TQFTs from non-semisimple modular categories

    [De +22] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. “3-Dimensional TQFTs from non-semisimple modular categories”.Sel. Math. New Ser.28.2 (2022),

    2022

  5. [5]

    Renormalized Hennings Invariants and 2+1- TQFTs

    [DGP18] M. De Renzi, N. Geer, and B. Patureau-Mirand. “Renormalized Hennings Invariants and 2+1- TQFTs”.Commun. Math. Phys.362.3 (2018), 855–907. [Eti+11] P. Etingof, O. Golberg, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina.Introduction to Representation Theory. Vol

    2018

  6. [6]

    Mapping class groups, skein algebras and combinatorial quantization

    [Fai19] M. Faitg. “Mapping class groups, skein algebras and combinatorial quantization”. PhD thesis. Universite de Montpellier, 2019.url:https://arxiv.org/abs/1910.04110. [Fai20] M. Faitg. “Projective Representations of Mapping Class Groups in Combinatorial Quantiza- tion”.Commun. Math. Phys.377.1 (2020), 161–198. [Fai24] M. Faitg. “Holonomy and (stated) ...

    work page arXiv 2019

  7. [7]

    Centrally extended mapping class groups from quantum Te- ichm¨ uller theory

    [FK14] L. Funar and R. Kashaev. “Centrally extended mapping class groups from quantum Te- ichm¨ uller theory”.Adv. Math.252 (2014), 260–291. [FM12] B. Farb and D. Margalit.A primer on mapping class groups. Princeton Mathematical Series. Princeton University Press,

    2014

  8. [8]

    A finite presentation of the mapping class group of a punctured surface

    [Ger01] S. Gervais. “A finite presentation of the mapping class group of a punctured surface”.Topology 40.4 (2001), 703–725. [Ger96] S. Gervais. “Presentation and central extensions of mapping class groups”.Trans. Amer. Math. Soc.348.8 (1996), 3097–3132. [GM11] P. M. Gilmer and G. Masbaum. “Maslov index, Lagrangians, Mapping Class Groups and TQFT”.Forum M...

    2001

  9. [9]

    Quantum Character Varieties

    arXiv:2501.02316. [Jor25] D. Jordan. “Quantum Character Varieties”.Encyclopedia of Mathematical Physics. 2nd ed. Elsevier, 2025, 635–647. [Kas95] C. Kassel.Quantum Groups. Graduate Texts in Mathematics

    work page arXiv 2025

  10. [10]

    The second homology groups of mapping class groups of orientable surfaces

    [KS03] M. Korkmaz and A. I. Stipsicz. “The second homology groups of mapping class groups of orientable surfaces”.Math. Proc. Cambridge Philos. Soc.134.3 (2003), 479–489. [Lyu95] V. V. Lyubashenko. “Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity”.Commun. Math. Phys.172.3 (1995), 467–51...

    2003

  11. [11]

    Die Signatur von Flachenb¨ undeln

    IRMA Lectures in Mathematics and Theoretical Physics. EMS Press, 2021, 109–130. [Mey64] W. Meyer. “Die Signatur von Flachenb¨ undeln”.Math. Ann.201 (1964), 269–264. [Mon93] S. Montgomery.Hopf Algebras and Their Actions on Rings. Vol

    2021

  12. [12]

    L’anomalie des representations des groupes modulaires des surfaces en quantifica- tion combinatoire

    [Mou] J. Moulai. “L’anomalie des representations des groupes modulaires des surfaces en quantifica- tion combinatoire”. In preparation. PhD thesis. Universite de Montpellier. [MR95] G. Masbaum and J. D. Roberts. “On central extensions of mapping class groups”.Math. Ann. 302.1 (1995), 131–150. [Rad12] D. E. Radford.Hopf algebras. Vol

    1995

  13. [13]

    Lagrangian mapping class groups from a group homological point of view

    [Sak12] T. Sakasai. “Lagrangian mapping class groups from a group homological point of view”.Al- gebraic Geom. Topol.12.1 (2012), 267–291. [Tur10] V. G. Turaev.Quantum invariants of knots and 3-manifolds. 2nd ed. De Gruyter Studies in Mathematics

    2012