REVIEW 4 major objections 2 minor 1 cited by
The second integral homology of even Artin groups
T0 review · 4 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For even Artin groups, the second integral homology is free abelian with an explicit commutator-power basis.
desk verdict Fatal flaw in Equation (2.4): halving exponents in Artin relations changes the group, so the main theorem is about the wrong objects. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine of the paper is Hopf's formula $H_2(F/R) = (R\cap[F,F])/[F,R]$, together with a presentation of the even Artin group in which every defining relator has the form $(a_i a_j)^{n(i,j)} (a_j a_i)^{-n(i,j)}$. Lemma 4.3 supplies the identity $(ab)^n(ba)^{-n} \equiv [a,b]^n$ modulo $[G,[G,G]]$, which converts relators into commutator-power classes. Independence is certified by Hall's theorem that the commutator images $[a_i,a_j]$ form a free-abelian basis of $[F,F]/[F,[F,F]]$. For the product computations, the cup-product formula uses Hopf's evaluation of $\varphi\smile\psi$ on commutator representatives, and the Pontryagin-product part uses the bar-resolution description $\langle g,h\rangle = [g|h]-[h|g]$.
What would settle it
In the free group on $a,b$, compute both sides of Lemma 4.3 for $n=2$: $(ab)^2(ba)^{-2}$ reduces to $[a,b]$, while the claimed congruence would force $[a,b]^{-1}$ into $[F,[F,F]]$; since that element is not a triple commutator, the representative identity behind Theorem 4.2 fails, and the basis statement would not follow from the given proof.
Extended reading notes
Core claim
The central statement is Theorem 4.2. Writing an even Artin group $A_M$ as $F/R$, with $F$ free on $a_1,\ldots,a_n$ and $R$ the normal closure of the relators $(a_i a_j)^{n(i,j)} (a_j a_i)^{-n(i,j)}$ for the finite entries of $M$, the theorem asserts that $H_2(A_M)$, identified with $R/[F,R]$ by Hopf's formula, is a free abelian group with basis $\{[a_i,a_j]^{n(i,j)} \bmod [F,R] : (i,j)\in B\}$. The proof turns each defining relator into a commutator-power representative via Lemma 4.3, and uses Hall's basis theorem to prove that those representatives are linearly independent. The subsequent results, Theorems 5.2, 6.3, 7.1, and 7.2, all follow from this explicit basis: cup products are computed by evaluating on the basis, Pontryagin products are identified with basis elements, and the Coxeter-group statements are obtained by mapping the Artin basis through the natural surjection to $W_M$.
Load-bearing premise
The load-bearing premise is the assertion in (2.4) that the even Artin relation with exponent $2n(i,j)$ is equivalent to the shorter relation with exponent $n(i,j)$ in the paper's alternating-word convention.
Editorial extensions
If this is right
- $H_2(A_M)$ is torsion-free and free abelian, with rank equal to the number of finite off-diagonal entries of the Coxeter matrix.
- The cup product on $H^1$ is completely determined: $\beta_i\smile\beta_j = 0$ for $i=j$ or $n(i,j)=\infty$, and $\beta_i\smile\beta_j = n(i,j)\beta_{ij}$ otherwise.
- Every basis class $\alpha_{ij}$ is the Pontryagin product $\langle a_i,(a_j a_i)^{2n(i,j)-1}\rangle$, so all classes in $H_2(A_M)$ are linear combinations of Pontryagin products.
- For every even Coxeter group $W_M$, $H_2(W_M)$ is an elementary abelian $2$-group with basis $\{[s_i,s_j]^{n(i,j)} \bmod [F',R']\}$, and the same Pontryagin-product representatives work there.
- The explicit $H_2$ basis gives an explicit dual basis of $H^2$, making the integral cohomology ring computable in degree 2 rather than only at the level of Betti numbers.
Reading between the lines
- If the shortened relations in (2.4) fail to define the same group as the full even-Artin relations, the theorem's basis would still describe the shortened group, but would not be the second homology of the standard even Artin group; a two-generator check with $m=4$ would settle this.
- Because the paper's proof never invokes the Salvetti complex, the same Hopf-formula route might be tried for other classes of Artin groups whose $K(\pi,1)$ status is unknown, as long as the defining relations can be rewritten as commutators.
- The explicit $H_2$ basis suggests computing higher integral cohomology operations such as Bockstein homomorphisms on even Artin groups, a step the paper does not take beyond degree-2 cup products.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to compute the second integral homology of even Artin groups using Hopf's formula. The main result, Theorem 4.2, asserts that H_2(A_M) is free abelian with an explicit basis consisting of the cosets [a_i,a_j]^{n(i,j)} modulo [F,R]. The paper then derives formulas for cup products (Theorem 5.2), expresses the homology basis via Pontryagin products (Theorem 6.3), and finally uses these results to give a basis for the second integral homology of even Coxeter groups (Theorem 7.1). The approach is purely algebraic and self-contained, but the central definition of an even Artin group is altered in Equation (2.4), so the groups analyzed are not the even Artin groups announced in the title.
Significance. If the main computation were correct, an explicit algebraic basis for H_2 of even Artin groups would be a useful and notable contribution, and the cup-product and Pontryagin-product applications would add to its value. The paper is self-contained, uses a classical Hopf-formula framework, and states very concrete formulas, which are all strengths. However, the main theorem is built on an incorrect simplification of the defining relations of even Artin groups, and the proof also contains an independent gap in a key lemma. As a result, the significance of the announced results is not realized for the intended class of groups.
major comments (4)
- [Section 2, Equation (2.4)] The simplification claimed in Equation (2.4) is false. Under the paper's own convention, (gh)^k denotes the alternating word of length k, so (a_i a_j)^{2n} is the square of (a_i a_j)^n as a group element, not a separate relation equivalent to (a_i a_j)^n = (a_j a_i)^n. For m(i,j)=4, the announced even Artin group has presentation <a,b | abab=baba>, which admits the nonabelian dihedral quotient D_8, whereas Equation (2.4) gives <a,b | ab=ba>, isomorphic to Z^2. Thus the groups studied in the paper are not even Artin groups, and Theorems 4.2, 5.2, 6.3, and 7.1 do not address the announced objects.
- [Section 4, proof of Theorem 4.2] The passage from Lemma 4.3 to the congruence modulo [F,R] is unjustified. Lemma 4.3 only provides a congruence modulo [F,[F,F]], and [F,[F,F]] is not generally contained in [F,R]. The sentence 'Since [R,R]⊂F and hence [[R,R],R]⊂[F,R]' does not establish the needed containment, so the replacement of the relator (a_i a_j)^{n(i,j)}(a_j a_i)^{-n(i,j)} by [a_i,a_j]^{n(i,j)} in R/[F,R] is not valid.
- [Section 4, Lemma 4.3] Lemma 4.3 fails already for n=2. In the free group on a,b, (ab)^2(ba)^{-2} = ab a^{-1}b^{-1} = [a,b], which is not congruent to [a,b]^2 modulo [F,[F,F]]; the asserted congruence would require w_2 = [a,b]^{-1}, and [a,b] is not an element of [F,[F,F]]. Hence the lemma's conclusion is false in general.
- [Section 4, Theorem 4.2] Even under the paper's own defining relations, Theorem 4.2 is false for a basic case. If n(i,j)=2 for a single pair (i,j), the presentation becomes <a,b | ab a^{-1}b^{-1}>, so the group is Z^2 and H_2 is infinite cyclic generated by [a,b] modulo [F,R]. The proposed basis element [a,b]^2 modulo [F,R] is twice a generator, not a basis. Thus the main theorem cannot be correct even for the modified groups introduced by Equation (2.4).
minor comments (2)
- [Section 1, Notation] The notation (gh)^m as an alternating word of length m conflicts with standard group exponentiation, where (gh)^m usually means m copies of gh. This notational clash appears central to the error in Equation (2.4) and should be addressed by a more standard or clearly separated notation.
- [Sections 5 and 6] There are several typographical errors, such as 'follwing', 'whch', 'essencially', 'beccause', and 'conisidered'. These do not affect the mathematics but should be corrected in any revision.
Circularity Check
No circularity: the paper's derivation is self-contained, though it contains a serious algebraic error in equation (2.4).
full rationale
The derivation is self-contained: Theorem 4.2 is proved directly from Hopf's formula, Lemma 3.1, and Hall's basis theorem for [F,F]/[F,[F,F]]; no parameter is fitted to data and no conclusion of the paper is fed back into the proof. The only self-citation is [1] (Akita-Liu), used in Section 7 to assert that the induced map rho_*:H2(A_M)->H2(W_M) is surjective and becomes an isomorphism after tensoring with Z/2. That is an external published theorem, not a premise invented for this paper, and it does not assume Theorem 4.2 or any other result proved here; it is independent support rather than a circular chain. Other cited results (Howlett, Hall, Hopf, Charney, Clancy-Ellis) are likewise used as external facts. There is, however, a serious correctness issue: equation (2.4) asserts that (a_i a_j)^{2n} = (a_j a_i)^{2n} implies (a_i a_j)^n = (a_j a_i)^n under the paper's own convention that (gh)^k is the alternating word of length k, which is false; and the proof of Theorem 4.2 passes from Lemma 4.3's congruence modulo [F,[F,F]] to the needed congruence modulo [F,R] without justification, since [F,[F,F]] is not contained in [F,R]. These are mathematical errors, not circularity: the announced conclusions are not equivalent to the inputs by construction, and they are false rather than tautological.
Assumptions & free parameters
assumptions (3)
- standard math Hopf's formula H_2(G) is isomorphic to R∩[F,F]/[F,R] for a presentation F/R.
- standard math Hall's theorem that [F,F]/[F,[F,F]] is free abelian with basis [a_i,a_j] for i<j.
- ad hoc to paper Equation (2.4): for even M, the Artin relation (a_i a_j)^{m(i,j)} = (a_j a_i)^{m(i,j)} simplifies to exponent n(i,j)=m(i,j)/2.
Cite this review
Pith. "Pith review of The second integral homology of even Artin groups." pith.science (2026). https://pith.science/paper/MQ2KRNBZ
@misc{pith2026250704577,
author = {Pith},
title = {Pith review of: The second integral homology of even Artin groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/MQ2KRNBZ}},
note = {Machine review of arXiv:2507.04577}
}
read the original abstract
In this paper, we compute the second integral homology of even Artin groups using Hopf's formula. We then apply our results to the computation of cup products and Pontryagin products on even Artin groups, as well as to the second integral homology of even Coxeter groups.
Forward citations
Cited by 1 Pith paper
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Groups admitting Wirtinger presentations and Gromov hyperbolic groups
Hyperbolic groups admitting twisted Wirtinger presentations have vanishing second rational homology, and vanishing second integral homology when torsion-free.
Reference graph
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