The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

Annie Holden

REVIEW 3 major objections 5 minor 14 references

For genus at least 3, the handlebody Torelli group and handlebody Johnson kernel map onto explicit Boolean-polynomial subspaces under the BCJ homomorphism; for genus at least 4, cup-product classes of order g^6 (for the Torelli group) and g

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2026-08-01 03:10 UTC pith:EJULRFYU

load-bearing objection New, clean, and likely correct: exact BCJ images for the handlebody Torelli subgroups, with the equality in Theorem A resting on a single cited theorem that needs checking and a computational core that is hard to verify in the posted text. the 3 major comments →

arxiv 2607.25191 v1 pith:EJULRFYU submitted 2026-07-28 math.GT math.AT

The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

Annie Holden This is my paper

classification math.GT math.AT MSC 57K20

keywords Birman–Craggs–Johnson homomorphismhandlebody Torelli grouphandlebody Johnson kernelBoolean polynomialsquadratic formssecond cohomologyabelian cyclesRochlin invariant

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

For genus at least 3, the paper determines exactly which elements of the Boolean polynomial space B_3 arise from the handlebody Torelli group HI_g^1 and its Johnson kernel HK_g^1 under the Birman–Craggs–Johnson (BCJ) homomorphism. The images are the explicit subspaces B^bi_3 and B^bi_2, spanned by monomials each containing at least one of the b_i homology classes that bound meridian disks in the fixed handlebody. The proof identifies σ(HI_g^1) with the smallest subspace generated by the BCJ value of a single bounding pair annulus twist under the handlebody group's symplectic action, and σ(HK_g^1) with its degree-2 slice. Because the BCJ homomorphism packages the Birman–Craggs invariants tied to the Rochlin invariant of homology spheres, the exact images give a concrete picture of how handlebody Torelli elements change Heegaard embeddings. For genus at least 4, the author then uses commuting twist pairs to show the induced cup product map on H^2 has image growing at least like g^6 for HI_g^1 and g^4 for HK_g^1, producing 2-torsion classes invisible to rational cohomology.

Core claim

The central claim is that the BCJ homomorphism, restricted to the handlebody Torelli group and its Johnson kernel, is surjective onto precisely defined subspaces of the Boolean polynomial algebra. Specifically, for g≥3, σ(HK_g^1)=B^bi_2 and σ(HI_g^1)=B^bi_3, where B^bi_2⊂B_2 and B^bi_3⊂B_3 consist of all square-free polynomials whose monomials contain at least one b_i, with the b_i chosen to be homology classes of curves bounding disks in the fixed handlebody. The proof has two load-bearing steps: the evaluation formulas expressing σ on separating disk twists and bounding pair annulus twists, and the fact that a single bounding pair annulus twist normally generates HI_g^1 in the handlebody g

What carries the argument

The central object is the Birman–Craggs–Johnson homomorphism σ from the Torelli group to the F_2-vector space B_3 of Boolean polynomial functions on quadratic forms over H_1(Σ_g^1; F_2). The key identity is the normal generation of the handlebody Torelli group: a single bounding pair annulus twist T_y T_{y'}^{-1} normally generates HI_g^1 inside the handlebody group, so σ(HI_g^1) is forced to be the smallest ℓℓSp_{2g}(F_2)-invariant subspace of B_3 containing σ(T_y T_{y'}^{-1}). The evaluation formulas for separating disk twists and bounding pair annulus twists convert these twists into sums of products of linear functions, allowing the paper to identify this subspace explicitly: it is B^bi_

Load-bearing premise

The equality in Theorem A rests on the claim that the BCJ image of the handlebody Torelli group is exactly the subspace generated by the BCJ value of a single bounding pair annulus twist under the handlebody group's symplectic action, which depends on the normal-generation theorem for that twist and on σ being equivariant; if that claim fails, B^bi_3 is only a lower bound for the true image.

What would settle it

For the explicit bounding pair described in the paper, compute the ℓℓSp_{2g}(F_2)-orbit of σ(T_y T_{y'}^{-1}) in B_3 and check whether it spans every monomial containing at least one b_i; if any such monomial is missing for some g≥3, Theorem A is false. Alternatively, evaluate σ on a generating set of separating disk twists and bounding pair annulus twists for HK_g^1 and HI_g^1 and verify that their images lie in B^bi_2/B^bi_3 and generate those subspaces.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

For g≥3, the images σ(HK_g^1) and σ(HI_g^1) are exactly B^bi_2 and B^bi_3, with dimensions (3g^2+g)/2 and (7g^3+5g)/6.
For g≥4, the image of σ*: H^2(B^bi_3; F_2)→H^2(HI_g^1; F_2) has dimension at least order g^6, and the same map for B^bi_2 has dimension at least order g^4 for both HI_g^1 and HK_g^1.
For g≥4, the integral lift of the BCJ homomorphism has image in H^2(HK_g^1; Z) of dimension at least order g^4.
The exact images imply there exist many Heegaard embeddings h and handlebody Torelli elements k for which the reglued homology sphere M(h,k) is not S^3 and has nontrivial Rochlin invariant.

Where Pith is reading between the lines

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

If the paper's conjecture that H^1(HI_g^1; F_2)≅B^bi_3 for large genus holds, the BCJ homomorphism would give a complete isomorphism, not just an image, describing all first F_2-cohomology of the handlebody Torelli group.
The index-matched wedge products are the only apparent obstruction to the lower bounds being sharp; a direct computation of whether those products vanish in H^2 would either strengthen Theorems B and C to exact dimensions or reveal new relations.
The same normal-generation-plus-evaluation strategy could be applied to other handlebody-defined subgroups, such as intersections of the handlebody group with higher Johnson kernels or level structures, to obtain exact BCJ images there.
The g=4 cases where certain basis types require distinct indices (noted in Remark 8.9) suggest small-genus behavior may differ; testing g=4 explicitly could show whether the order-of-growth estimates are attained uniformly.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

New, clean, and likely correct: exact BCJ images for the handlebody Torelli subgroups, with the equality in Theorem A resting on a single cited theorem that needs checking and a computational core that is hard to verify in the posted text.

read the letter

Two things to know up front. The main results are new: the first exact computation of the BCJ images for the handlebody Torelli group and the handlebody Johnson kernel, plus cup-product lower bounds of order g^6 and g^4 modeled on Brendle–Farb. The one load-bearing external input is a normal-generation theorem of Omori that she states but does not reproduce. If that theorem says what she says it does, Theorem A is proved, because the two directions bracket each other.

The strategy for Theorem A is clean. Lemma 5.1 shows B^bi_3 sits inside sigma(HI_g^1) by explicit evaluation on separating twists and bounding pair twists, then applying the handlebody symplectic generators. Lemma 5.2 shows B^bi_3 is invariant and contains sigma of the one twist. Given Omori's normal generation, sigma(HI_g^1) is exactly the orbit-span of that single element, so the reverse inclusion follows; the two containments close. No circularity, no fitted parameters, and the dimension counts in Proposition 5.6 are consistent with the stated formulas.

Soft spots, in proportion. The equality in Theorem A genuinely rests on [11, Thm 1.2]. She does state the needed version explicitly — one bounding pair annulus twist normally generates HI_g^1 in H_g^1 — so the paper is internally coherent. But the referee should verify Omori's theorem actually covers that ambient group, that genus range, and that twist. If it doesn't, the reverse containment collapses and Theorem A becomes a lower bound only. That is a citation-accuracy check, not a demonstrated flaw, but it is the crux. Second, the proofs of Theorems 7.1 and 8.1 are long enumerations of wedge-product types, and the posted text is corrupted with interleaved garbage tokens that make them painful to read. The dimension counts (Propositions 7.2 and 8.14) are readable and plausible, and since Theorems B and C are lower bounds, a missed type would weaken but not necessarily kill them; still, the "all non-index-matched elements" claims deserve a patient checker. Third, Conjecture 1.1 is labeled a conjecture — no overselling.

Who gets value: people in mapping class groups, handlebody groups, and Rochlin invariant land. It deserves a serious referee. I would send it out and ask specifically for two things: verify the Omori quotation, and spot-check the enumeration in Section 8 against a clean version of the text.

Referee Report

3 major / 5 minor

Summary. The paper studies the Birman–Craggs–Johnson (BCJ) homomorphism σ on the handlebody Torelli group HI_g^1 and the handlebody Johnson kernel HK_g^1. It defines explicit subspaces B^bi_2 ⊂ B_2 and B^bi_3 ⊂ B_3 of Boolean polynomial spaces, and claims (Theorem A) that for g ≥ 3, σ(HK_g^1) = B^bi_2 and σ(HI_g^1) = B^bi_3. The paper then uses abelian cycles from commuting separating disk twists and bounding pair annulus twists to show (Theorems B and C) that the induced maps on second cohomology have image of dimension at least order g^6 and g^4, respectively, and Corollary 1.2 lifts the HK result to integral coefficients via Morita's homomorphism. The overall strategy is containment, invariance under ℓℓSp_{2g}(F_2), and normal generation for HI, combined with explicit orbit computations and dimension counts.

Significance. If Theorem A is correct, it gives a complete determination of the F_2-image of the BCJ homomorphism on the handlebody Torelli group and handlebody Johnson kernel, thereby identifying H^1(σ(HI)) and H^1(σ(HK)) with explicit subspaces of H^1 of the corresponding groups. The subsequent cup-product and abelian-cycle computations provide many new torsion classes in H^2 that are not rationally detectable, directly extending the Brendle–Farb framework. The paper is commendable for its explicit, parameter-free combinatorial definitions of B^bi_2 and B^bi_3 and for the concrete dimension counts, which make the main assertions falsifiable and easy to check in principle.

major comments (3)

[§5.3, proof of Theorem A] The equality σ(HK_g^1) = B^bi_2 is not established by the cited lemmas. Lemma 5.1 proves only the containment B^bi_2 ⊂ σ(HK_g^1). Lemma 5.2 concerns B^bi_3 and gives no normal-generation statement for HK_g^1; it only shows B^bi_3 is an ℓℓSp_{2g}(F_2)-subspace containing σ(T_yT_{y'}^{-1}). No argument or reference is supplied for the reverse containment σ(HK_g^1) ⊂ B^bi_2. As written, the first bullet of Theorem A could be only a lower bound. The authors must provide a proof of the missing containment or a precise reference (e.g., a normal generation statement for HK_g^1 in H_g^1 by the relevant twists).
[§5.3, Theorem A (HI equality)] The reverse containment for HI rests entirely on the assertion that the single bounding pair annulus twist T_yT_{y'}^{-1} normally generates HI_g^1 inside H_g^1, quoting [11, Theorem 1.2]. The statement of that theorem is not reproduced, so the referee cannot verify the ambient group, the genus range, or the exact twist. If the theorem gives normal generation in Mod_g^1 rather than H_g^1, the invariant subspace under Sp_{2g}(F_2) could be strictly larger than B^bi_3, invalidating the equality. Please state the theorem verbatim and confirm that it applies as used.
[§7.1 and §8.1–8.13] The proofs of Theorem 7.1, Theorem 8.1, and the supporting Propositions 8.2–8.13 contain large passages of corrupted, unreadable typesetting—long strings such as '⌟⟨⟨⟪rl⟫l⟩⟩...' are interleaved with equations, and many displayed computations are incomplete or mislabeled. As printed, these proofs cannot be verified. Since Theorems B and C are central claims, the authors must rewrite these sections completely, with every claimed action computed explicitly and legibly. This is not merely a cosmetic issue.

minor comments (5)

[§7] Theorem C states the result for G = HI_g^1 or HK_g^1, but Section 7 only constructs classes in H_2(HK_g^1). The paper should explicitly note that these classes push forward to H_2(HI_g^1) under inclusion and that naturality of σ_* transfers the lower bound to HI_g^1.
[§7.1] The definition of 'index-matched' and the accompanying footnote 6 are confusing, especially the sentence 'We allow ¯y=¯bi to include elements of the form ¯ai ¯x∧¯bi ¯y.' Please clarify the allowed choices for x and y and the convention for b_i^2.
[§6.2] The paper states that for an abelian cycle, σ_*({f,h}) = (σ(f)∧σ(h),0) in H_2(B;F_2) ≅ ⋀^2 B ⊕ B, but it does not explain why the Tor term component always vanishes. While this follows from [2], a sentence of explanation would improve readability.
[§5.5] In Proposition 5.6, the phrase 'B_r has ∑_{i=0}^r (g choose i) more generators than B^bi_r' is awkward; it should be phrased as 'B_r has ... additional generators.'
[§5.2] Proposition 5.5 contains a typo: '31≤i,j≤g' should read '1≤i,j≤g.'

Circularity Check

0 steps flagged

No significant circularity: the B^bi subspaces are defined combinatorially and the reverse containments rest on external normal-generation and Johnson theorems, not on the paper's own conclusions.

full rationale

The central claim Theorem A equates sigma(HK_g^1) and sigma(HI_g^1) with explicitly defined combinatorial subspaces B^bi_2 and B^bi_3. Those subspaces are defined in Section 5 purely from the symplectic basis (a_i,b_i), before any appeal to sigma, and are independent of the images being computed. Lemma 5.1 proves containment B^bi_r ⊂ sigma(...) using Johnson's evaluation formulas (Lemmas 5.3, 5.4) and the H_g^1-equivariance of sigma. Lemma 5.2 proves B^bi_3 is an ℓℓSp_{2g}(F2)-invariant subspace containing sigma(T_yT_y'^{-1}). The reverse containment uses the cited theorem of Omori [11, Theorem 1.2] that a single bounding pair twist normally generates HI_g^1 in H_g^1, together with equivariance; this is an external input, not a result of the present paper. No parameter is fitted and no predicted quantity is equivalent by construction to an input. The only self-citation, [6], is used for background and comparison ('our methods did not yield...information about torsion...'), not to prove Theorem A, B, or C. The lower-bound cohomology theorems B and C are computed from external formulas of Johnson and Brendle–Farb, and their dimension counts are purely combinatorial. Thus there is no circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claim rests on standard mapping-class-group background and on three external theorems (Johnson, Hirose, Omori). No new physical or mathematical entities are postulated; B^bi_2 and B^bi_3 are defined subspaces of existing spaces. No free parameters are fit to data.

axioms (5)

standard math Johnson's BCJ homomorphism σ: I_g^1 → B_3 exists, is surjective, and σ(K_g^1) = B_2.
Used throughout; the paper cites [8] and does not re-derive these foundational facts.
domain assumption Johnson's evaluation formulas: a separating disk twist T_x maps to ∑ c_i d_i, and a bounding pair annulus twist T_y T_{y'}^{-1} maps to (∑ c_i d_i)([y]+1).
This is the computational engine for Lemma 5.1 and for the abelian-cycle computations; cited as [8, Lemmas 12a, 12b].
domain assumption Hirose's theorem: the image of the handlebody group in Sp_{2g}(Z) is ℓℓSp_{2g}(Z) = { [A 0; B (A^t)^{-1}] }.
Used in Section 4.1 and to identify the ℓℓSp_{2g}(F2) action under which σ images are closed.
domain assumption Omori's theorem: a single bounding pair annulus twist T_y T_{y'}^{-1} normally generates HI_g^1 in H_g^1.
This is the key input that turns containment B^bi_3 ⊂ σ(HI_g^1) into equality via invariance; cited as [11, Theorem 1.2].
standard math Sakasai's abelian-cycle formula: σ_*({f,h}) = (σ(f) ∧ σ(h), 0) in H_2(B; F2).
Used to convert commuting elements of G into elements of im(σ_*); cited as [13, Lemma 2.1].

pith-pipeline@v1.3.0-alltime-deepseek · 114604 in / 21914 out tokens · 207568 ms · 2026-08-01T03:10:07.358101+00:00 · methodology

0 comments

read the original abstract

We use the Birman--Craggs--Johnson (BCJ) homomorphism to study the intersection of the handlebody group with the Torelli group and with the Johnson kernel. For genus $\geq 3$, we determine the images of the BCJ homomorphism restricted to these subgroups. We then explicitly compute cup products of pairs of cohomology classes in $H^1(-;\mathbb{F}_2)$ detected by the BCJ homomorphism for genus $\geq 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.

Figures

Figures reproduced from arXiv: 2607.25191 by Annie Holden.

**Figure 2.** Figure 2: A separating disk twist Tx and a bounding pair annulus twist TyT −1 y′ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Commuting disk twists Tx and Ty The element σ∗({Tx, Ty}) = a¯1 ¯b1 ∧ a¯2 ¯b2 is Type 1. We now find the rest of the basis element types in im(σ∗), building on previous work as we go. We have Xjk(a¯i ¯bi ∧ a¯j ¯bj ´ ¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¶ Type 1 ) = a¯i ¯bi ∧ a¯j ¯bj + a¯i ¯bi ∧ a¯j ¯bk ´ ¹¹¹¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Type 2 Yjℓ(a¯i ¯bi ∧ a¯ℓ ¯bk ´ ¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¶ Type 2 ) = a¯i ¯bi ∧ a¯ℓ ¯bk + a¯i… view at source ↗

**Figure 4.** Figure 4: Disk twists and bounding pair annulus twists where a¯1 ¯b1 ∧ a¯3 ¯b3a¯4 ∈ im(σ∗) is Type 7. We now show that the remaining element types lie in im(σ∗). We have Xjk(a¯i ¯bi ∧ a¯ia¯j ¯bj ´ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Type 2 ) = a¯i ¯bi ∧ a¯ia¯j ¯bj + a¯i ¯bi ∧ a¯ia¯j ¯bk ´ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Type 3 . We also have Xki(a¯i ¯bi ∧ a¯ja¯k ¯bk ´ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Type 7 ) = a¯i … view at source ↗

**Figure 5.** Figure 5: Bounding pair annulus twists TxT −1 x′ and TyT −1 y′ For some choice of indexing, we have σ∗({TxT −1 x′ , TyT −1 y ′ }) = a¯ia¯j ¯bi ∧ a¯ka¯ℓ ¯bk ´ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Type 4 + LOTs [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 4 linked inside Pith

[1]

Birman & R

J. Birman & R. Craggs, The µ-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold, Trans. Amer. Math. Soc. 237 (1978), 283–309. (Cited on page 4.)

1978
[2]

Brendle & B

T. Brendle & B. Farb, The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group, Math. Ann. 338 (2007), no. 1, 33–53. arXiv:math/0601163 (Cited on pages 1, 2, 3, 11, and 12.)

Pith/arXiv arXiv 2007
[3]

Eells & N

J. Eells & N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. (Cited on page 1.)

1962
[4]

Hensel, A primer on handlebody groups, Handbook of group actions (Vol

S. Hensel, A primer on handlebody groups, Handbook of group actions (Vol. V), ALM 48, Ch. 4 pp. 143–177 (2020). (Cited on page 1.)

2020
[5]

Hirose, The action of the handlebody group on the first homology group of the surface, Kyungpook Math

S. Hirose, The action of the handlebody group on the first homology group of the surface, Kyungpook Math. J. 46 (3) (2006) 399–408. (Cited on page 5.)

2006
[6]

Holden, Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups, preprint (2025).arXiv:2509.03742(Cited on pages 1 and 2.)

A. Holden, Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups, preprint (2025).arXiv:2509.03742(Cited on pages 1 and 2.)

Pith/arXiv arXiv 2025
[7]

Johnson, The structure of the Torelli group

D. Johnson, The structure of the Torelli group. III. The abelianization ofT, Topology 24 (1985), no. 2, 127–144. (Cited on page 1.) 10We account for ¯bi ∉{¯w,¯z}and remove{¯ai,¯y}as a choice. 23

1985
[8]

Johnson, Quadratic forms and the Birman-Craggs homomorphisms, Trans

D. Johnson, Quadratic forms and the Birman-Craggs homomorphisms, Trans. Amer. Math. Soc. 261 (1980), no. 1, 235–254. (Cited on pages 1, 4, 5, 6, 7, and 11.)

1980
[9]

Minahan & A

D. Minahan & A. Putman, The second rational homology of the Torelli group, preprint (2025). arXiv:2504.00211(Cited on page 1.)

Pith/arXiv arXiv 2025
[10]

Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, 28 (1989), 305-323

S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, 28 (1989), 305-323. (Cited on page 3.)

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[11]

Omori, A small normal generating set for the handlebody subgroup of the Torelli group, Geometriae Dedicata, 201(1):353-367 (2019).arXiv:1607.06553(Cited on page 6.)

G. Omori, A small normal generating set for the handlebody subgroup of the Torelli group, Geometriae Dedicata, 201(1):353-367 (2019).arXiv:1607.06553(Cited on page 6.)

Pith/arXiv arXiv 2019
[12]

O. T. O’Meara,Symplectic Groups, American Mathematical Society, Providence (1978). (Cited on page 7.)

1978
[13]

Sakasai, The Johnson homomorphism and the third rational cohomology group of the Torelli group, Topology Appl., 148 (2005), 83-111

T. Sakasai, The Johnson homomorphism and the third rational cohomology group of the Torelli group, Topology Appl., 148 (2005), 83-111. (Cited on page 12.)

2005
[14]

Suzuki, On homeomorphisms of a 3-dimensional handlebody, Can

S. Suzuki, On homeomorphisms of a 3-dimensional handlebody, Can. J. Math. 29 (1977), 111-124. (Cited on page 7.) Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Email address:holdena@sas.upenn.edu

1977

[1] [1]

Birman & R

J. Birman & R. Craggs, The µ-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold, Trans. Amer. Math. Soc. 237 (1978), 283–309. (Cited on page 4.)

1978

[2] [2]

Brendle & B

T. Brendle & B. Farb, The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group, Math. Ann. 338 (2007), no. 1, 33–53. arXiv:math/0601163 (Cited on pages 1, 2, 3, 11, and 12.)

Pith/arXiv arXiv 2007

[3] [3]

Eells & N

J. Eells & N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. (Cited on page 1.)

1962

[4] [4]

Hensel, A primer on handlebody groups, Handbook of group actions (Vol

S. Hensel, A primer on handlebody groups, Handbook of group actions (Vol. V), ALM 48, Ch. 4 pp. 143–177 (2020). (Cited on page 1.)

2020

[5] [5]

Hirose, The action of the handlebody group on the first homology group of the surface, Kyungpook Math

S. Hirose, The action of the handlebody group on the first homology group of the surface, Kyungpook Math. J. 46 (3) (2006) 399–408. (Cited on page 5.)

2006

[6] [6]

Holden, Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups, preprint (2025).arXiv:2509.03742(Cited on pages 1 and 2.)

A. Holden, Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups, preprint (2025).arXiv:2509.03742(Cited on pages 1 and 2.)

Pith/arXiv arXiv 2025

[7] [7]

Johnson, The structure of the Torelli group

D. Johnson, The structure of the Torelli group. III. The abelianization ofT, Topology 24 (1985), no. 2, 127–144. (Cited on page 1.) 10We account for ¯bi ∉{¯w,¯z}and remove{¯ai,¯y}as a choice. 23

1985

[8] [8]

Johnson, Quadratic forms and the Birman-Craggs homomorphisms, Trans

D. Johnson, Quadratic forms and the Birman-Craggs homomorphisms, Trans. Amer. Math. Soc. 261 (1980), no. 1, 235–254. (Cited on pages 1, 4, 5, 6, 7, and 11.)

1980

[9] [9]

Minahan & A

D. Minahan & A. Putman, The second rational homology of the Torelli group, preprint (2025). arXiv:2504.00211(Cited on page 1.)

Pith/arXiv arXiv 2025

[10] [10]

Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, 28 (1989), 305-323

S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, 28 (1989), 305-323. (Cited on page 3.)

1989

[11] [11]

Omori, A small normal generating set for the handlebody subgroup of the Torelli group, Geometriae Dedicata, 201(1):353-367 (2019).arXiv:1607.06553(Cited on page 6.)

G. Omori, A small normal generating set for the handlebody subgroup of the Torelli group, Geometriae Dedicata, 201(1):353-367 (2019).arXiv:1607.06553(Cited on page 6.)

Pith/arXiv arXiv 2019

[12] [12]

O. T. O’Meara,Symplectic Groups, American Mathematical Society, Providence (1978). (Cited on page 7.)

1978

[13] [13]

Sakasai, The Johnson homomorphism and the third rational cohomology group of the Torelli group, Topology Appl., 148 (2005), 83-111

T. Sakasai, The Johnson homomorphism and the third rational cohomology group of the Torelli group, Topology Appl., 148 (2005), 83-111. (Cited on page 12.)

2005

[14] [14]

Suzuki, On homeomorphisms of a 3-dimensional handlebody, Can

S. Suzuki, On homeomorphisms of a 3-dimensional handlebody, Can. J. Math. 29 (1977), 111-124. (Cited on page 7.) Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Email address:holdena@sas.upenn.edu

1977