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

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Pith's one-line read The stated skein algebra of a surface paired with a factorizable ribbon Hopf algebra determines an explicit central extension of the mapping class group.

desk verdict The paper computes the central extension of the mapping class group from stated skein algebras via a direct 2D argument for surfaces with at most one boundary. read the letter →

arxiv 2606.24378 v1 pith:7C3SQTXN submitted 2026-06-23 math.QA math.GT

Joris Moulai This is my paper · ORCID
classification math.QAmath.GT
keywords mappingclassgroupscentralextensionsstatedskeinalgebrasribbonHopfprojectiverepresentationssurfacesquantumtopology
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The pith

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

The reading

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.

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.

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.

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Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 0 assumptions · 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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Pith. "Pith review of Central extensions of mapping class groups of surfaces from stated skein algebras." pith.science (2026). https://pith.science/paper/7C3SQTXN

@misc{pith2026260624378,
  author       = {Pith},
  title        = {Pith review of: Central extensions of mapping class groups of surfaces from stated skein algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7C3SQTXN}},
  note         = {Machine review of arXiv:2606.24378}
}
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.

Figures

Figures reproduced from arXiv: 2606.24378 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
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. 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 ↗
Figures from the paper (15 more)
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 simp…
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]
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 ⊗ · · · …
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
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 …
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png]
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 …
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 −−−…
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png]
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+ ∈ …
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png]
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…
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p041_2.png]

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