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arxiv: 2607.02107 · v1 · pith:GHISXSHYnew · submitted 2026-07-02 · 🧮 math.GT · math.GR

Torsion in the homology of the Torelli group and the Birman-Craggs-Johnson homomorphism

Andrei Vladimirov This is my paper

Pith reviewed 2026-07-03 02:53 UTC · model grok-4.3

classification 🧮 math.GT math.GR
keywords Torelli groupBirman-Craggs-Johnson homomorphismabelian cyclesseparating Dehn twistshomology of mapping class groupsinjectivity on homology
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The pith

The Birman-Craggs-Johnson homomorphism injects the subgroup of abelian cycles in H_k of the Torelli group when k is at most g-2.

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

The paper extends a 1983 result of Johnson from the first homology of the Torelli group to higher degrees. It considers abelian cycles in H_k(I_g) generated by Dehn twists about pairwise disjoint separating simple closed curves, which commute and therefore produce well-defined homology classes. The central result is that the map on homology induced by the Birman-Craggs-Johnson homomorphism remains injective when restricted to the subgroup generated by these cycles, provided k ≤ g-2. A reader would care because the Torelli group sits inside the mapping class group and its homology controls many questions about surface diffeomorphisms; an explicit homomorphism that detects nontrivial classes supplies a concrete way to produce lower bounds in a range where direct calculation is difficult.

Core claim

Given any collection of pairwise disjoint separating simple closed curves on a surface of genus g, the corresponding Dehn twists determine an abelian cycle in H_k(I_g). The induced homomorphism on homology coming from the Birman-Craggs-Johnson map σ : I_g → B_3' is injective on the subgroup generated by all such cycles whenever k ≤ g-2.

What carries the argument

The pushforward map induced by the Birman-Craggs-Johnson homomorphism on the subgroups of H_k(I_g) generated by abelian cycles of disjoint separating Dehn twists.

If this is right

  • The abelian cycles remain linearly independent in H_k(I_g) for k ≤ g-2.
  • The result supplies a lower bound on the dimension of the image of these cycles inside the homology with Z/2Z coefficients.
  • The construction recovers Johnson's original injectivity statement when k=1.
  • The same cycles can be used to detect nontrivial torsion in the homology groups in the stated range.

Where Pith is reading between the lines

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

  • The range k ≤ g-2 may be the stable range in which the BCJ map continues to detect the full span of these cycles.
  • Similar injectivity statements could be tested for other natural maps out of the Torelli group once their effect on separating twists is known.
  • The result suggests that the torsion detected by these cycles persists in the homology of the full mapping class group in the same range.

Load-bearing premise

The Birman-Craggs-Johnson homomorphism is a well-defined group homomorphism whose induced map on homology sends the chosen abelian cycles to linearly independent elements in the target vector space.

What would settle it

An explicit collection of k ≤ g-2 pairwise disjoint separating curves on Σ_g such that the corresponding abelian cycle lies in the kernel of the induced BCJ map on homology while being nonzero in H_k(I_g).

Figures

Figures reproduced from arXiv: 2607.02107 by Andrei Vladimirov.

Figure 1
Figure 1. Figure 1: FiGURE 1. Surfaces [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FiGURE 2. Subsurface [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FiGURE 3. Example of the curves [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

The Birman-Craggs-Johnson homomorphism is a homomorphism $\sigma \colon \mathcal{I}_g \to \mathbb{B}_3'$ from the Torelli group to a certain $\mathbb{Z}/2\mathbb{Z}$-vector space of Boolean polynomials. In 1983, Johnson computed $H_1(\mathcal{I}_g)$ for $g \geq 3$ and showed, in particular, that the induced homomorphism on $H_1(\mathcal{I}_g)$ is injective when restricted to the subgroup generated by Dehn twists about separating simple closed curves. In this paper, we extend Johnson's result to higher homology groups. Given any collection of pairwise disjoint separating simple closed curves on $\Sigma_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$ called an abelian cycle. We prove that the pushforward homomorphism restricted to the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is injective for $k \leq g-2$.

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

1 major / 2 minor

Summary. The paper extends Johnson's 1983 computation of H_1(I_g) by proving that the Birman-Craggs-Johnson homomorphism σ: I_g → B_3' induces an injection on the F_2-span of abelian cycles in H_k(I_g) generated by pairwise disjoint separating Dehn twists, for all k ≤ g-2.

Significance. If the result holds, it supplies new structural information on torsion in the homology of the Torelli group in degrees up to roughly g-2, extending a classical computation in a natural direction and potentially aiding computations of stable homology or related invariants in mapping class groups.

major comments (1)
  1. [main theorem / §4 (proof of injectivity for k>1)] The central injectivity claim for k > 1 rests on showing that the images of these abelian cycles remain linearly independent in H_k(B_3'). The manuscript must supply an explicit argument (e.g., in the section containing the main theorem) that the Boolean-polynomial construction introduces no additional kernel elements when k commuting twists are multiplied in the group ring before passing to homology; the k=1 case from Johnson does not automatically extend.
minor comments (2)
  1. [Introduction / §2] Notation for the target space B_3' and the precise definition of the Boolean polynomials should be recalled or referenced at the start of the higher-homology argument for readability.
  2. [Abstract / Theorem statement] The range k ≤ g-2 is stated without an accompanying remark on whether the bound is sharp or merely convenient; a brief comment on this would clarify the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point in the proof of the main theorem that would benefit from greater explicitness. We address the major comment below.

read point-by-point responses

  1. Referee: [main theorem / §4 (proof of injectivity for k>1)] The central injectivity claim for k > 1 rests on showing that the images of these abelian cycles remain linearly independent in H_k(B_3'). The manuscript must supply an explicit argument (e.g., in the section containing the main theorem) that the Boolean-polynomial construction introduces no additional kernel elements when k commuting twists are multiplied in the group ring before passing to homology; the k=1 case from Johnson does not automatically extend.

    Authors: We agree that the linear independence in H_k(B_3') for k>1 requires an explicit verification that the Boolean-polynomial images of products of k commuting separating twists introduce no extraneous kernel elements beyond the k=1 case. In the revised manuscript we will insert a self-contained paragraph (or short subsection) immediately preceding the statement of the main theorem that carries out this verification: because σ is a group homomorphism and the target is an F_2-vector space whose multiplication is given by symmetric Boolean polynomials, the image of an abelian cycle is the wedge product of the individual images; the explicit form of σ on separating twists (as recorded in Johnson’s original work and extended by the Boolean-polynomial definition) ensures that these wedge products remain linearly independent precisely when the curves are pairwise disjoint and k ≤ g−2. This argument uses only the already-established properties of σ and does not rely on any new computations. revision: yes

Circularity Check

0 steps flagged

No circularity: extension of external Johnson 1983 result on independent prior computation

full rationale

The derivation extends Johnson's 1983 explicit computation of H_1(I_g) and injectivity on separating twists to higher k via the Birman-Craggs-Johnson map σ. The abstract and claim rely on the standard well-definedness of σ and commutativity of disjoint Dehn twists (external facts from mapping class group theory), without any self-citation, self-definition of the target space, or reduction of the higher-homology injectivity statement to a fitted parameter or prior result by the same author. The central statement for k ≤ g-2 is presented as new content building on the k=1 case, with no load-bearing step that collapses to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only. The claim rests on the prior existence of the Birman-Craggs-Johnson homomorphism and standard facts about Dehn twists and homology of groups; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Birman-Craggs-Johnson homomorphism is a well-defined homomorphism I_g → B_3' that induces maps on homology.
    Invoked in the abstract as the object whose pushforward is studied.
  • domain assumption Pairwise disjoint separating simple closed curves give commuting Dehn twists whose homology classes form abelian cycles.
    Stated directly in the abstract as the generators under consideration.

pith-pipeline@v0.9.1-grok · 5712 in / 1427 out tokens · 39410 ms · 2026-07-03T02:53:47.221649+00:00 · methodology

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Reference graph

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15 extracted references · 3 canonical work pages · 2 internal anchors

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