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Homological properties of parafree Lie algebras

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Pith's one-line read Explicit parafree Lie algebra has nonzero second homology

desk verdict Two of the three main results hold up; Theorem B's proof has a real gap in how it treats the completed exterior square. read the letter →

arxiv 1908.04608 v2 pith:ITZMPH4X submitted 2019-08-13 math.RA math.GR

Sergei O. Ivanov , Roman Mikhailov , Anatoly Zaikovski This is my paper · ORCID
classification math.RAmath.GR MSC 17B5517B0117B65
keywords parafreeLiealgebrasecondhomologypronilpotentcompletioncohomologicaldimensionformalpowerseriesnon-rationallamplighterStallingstheorem
verification ladder T0 review T1 audit T2 compute T3 formal

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 constructs an explicit countable parafree Lie algebra with nonzero second homology, the first such example given by generators and relations. Over a field K of characteristic two, the quotient a = b/γω(b) of the explicitly presented algebra b is parafree, and it carries a nonzero class in H2(a,K). The same method shows that the pronilpotent completion of the free Lie algebra on two generators over Z has cohomological dimension greater than two, because H2 of that completion contains a nonzero element divisible by every power of two. The paper also proves that the pronilpotent completion of a free group of rank at least two contains uncountably many countable parafree subgroups with nonzero H2. A sympathetic reader should care because it pins down where the Parafree Conjecture can hold: finite generation is essential, and explicit homological counterexamples exist just beyond it.

What carries the argument

The central object is the Lie-algebra analogue of the lamplighter group: l_R = R[x] ⋊ Q, where Q is a one-dimensional free module R t and the generator t acts on polynomials by [p,t] = px; its pronilpotent completion is R[[x]] ⋊ Q with [f,t] = fx. Working through the Lyndon–Hochschild–Serre spectral sequence, the paper identifies H2(\hat l_R, R) with the tensor product R[[x]] \wedge_{R[x]} R[[x]] over the polynomial ring (Lemmas 2.6–2.7), and shows that a class p ∧ 1 is nonzero whenever p is a formal power series that is not a rational function (Lemmas 2.8–2.9). The cycles feeding this detection are produced in the free Lie algebra f_R(a,b) by the Engel-type identity [ [x, 2^n y], x] = [ \sum_{i=0}^{n-1} (-1)^i [[x, $2^{{n-1-i}}$ y], [x, 2^i y]], y ], which makes r_{2^n} = [a, 2^n b] and s_{2^n} = \sum_{i=1}^n (-1)^i [[a, $2^{{n-i}}$ b], [a, $2^{{i-1}}$ b]] satisfy [r_{2^n}, a] + [s_{2^n}, b] = 0; hence weighted infinite sums R_α ∧ a + S_α ∧ b are cycles in \hat f \tilde\wedge \hat f. Their images in H2(\hat l_R) are non-rational series p_α ∧ 1 = (\sum α_n $x^{{2^n}}$) ∧ 1, establishing uncountability in Theorem 2.11, and the power series \sum 2^n $x^{{2^n}}$ ∧ 1 produces the 2-divisible class for Theorem B. For Theorem A, a characteristic-2 identity, [a, $2^{{n+1}}$ b, a] = [a, 2^n b, a, 2^n b], converts the defining relations of b into the same non-rationality argument and yields x_1 ∧ a + y_1 ∧ b as the nonzero class.

What would settle it

Compute inside the completed exterior square \hat f \tilde\wedge \hat f and check whether the partial sums \sum_{n=1}^N 2^n (r_{2^n} ∧ a + s_{2^n} ∧ b) form a Cauchy sequence for any pronilpotent topology; if in the limit the sum is zero because the filtration of the exterior square is not Hausdorff, then the claimed 2-divisible class in H2(\hat f, Z) vanishes and Theorem B collapses.

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Extended reading notes

Core claim

Over a field K of characteristic 2, let b be the Lie algebra generated by a, b, and two countable families {x_i}_{i≥1}, {y_i}_{i≥1} subject to x_i = [a,b,b] + [x_{i+1}, 2^i b] and y_i = [a,b,a] + [y_{i+1}, 2^i b] for all i ≥ 1. The paper proves that a = b/γω(b), where γω(b) is the intersection of the lower central series, is parafree: the map from the free Lie algebra on {a,b} sending the generators to a and b induces isomorphisms on every lower central quotient. It then shows H2(a,K) ≠ 0 by exhibiting the cycle x_1 ∧ a + y_1 ∧ b and proving its image in the homology of a related lamplighter Lie algebra is nonzero. For the integral case, writing f for the free Lie algebra on two generators over Z and \hat f for its pronilpotent completion, the paper proves that \sum_{n≥1} 2^n (r_{2^n} ∧ a + s_{2^n} ∧ b) defines a nonzero element of H2(\hat f, Z), that this element is 2^k-divisible for every k, and consequently that the cohomological dimension of \hat f is at least 3. Finally, it shows that the pronilpotent completion of a noncyclic finitely generated free group is the filtered union of its countable parafree subgroups, and since H2 commutes with filtered colimits, uncountably many of those countable parafree groups have nonzero second homology.

Load-bearing premise

The load-bearing premise is that infinite linear combinations such as \sum α_n (r_{2^n} ∧ a + s_{2^n} ∧ b) and \sum 2^n (r_{2^n} ∧ a + s_{2^n} ∧ b) are legitimate elements of the exterior square of the pronilpotent completion, and that H2 of the completion still equals the kernel of the wedge map; the paper does not define the topology or completed exterior square making these sums converge.

Editorial extensions

If this is right

  • The Parafree Conjecture as usually stated for finitely generated groups has no direct analogue for countably generated Lie algebras: Theorem A supplies an explicit countable parafree Lie algebra with H2 ≠ 0.
  • The pronilpotent completion of a free Lie algebra of rank at least two over Z has cohomological dimension at least three, so Stallings–Swan-type dimension-one rigidity fails in the category of pronilpotent Lie algebras.
  • H2(\hat f, Z) contains a nonzero element divisible by every power of 2, so it is not a free abelian group; consequently the trivial module over U(\hat f) has projective dimension at least three.
  • There are uncountably many countable parafree subgroups of the pronilpotent completion of a free group with nonzero H2, showing the countable parafree landscape is large rather than a single pathology.
  • The explicit cycles are detected by formal power series that are not rational, so any parafree Lie algebra admitting a similar map to the lamplighter Lie algebra with non-rational image will again have nonzero H2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same non-rationality technique should produce finitely presented parafree Lie algebras with nonzero H2 over Z, not just countably presented ones: characteristic 2 is convenient for the Engel identities, but the 2-divisibility argument of Theorem B is characteristic-free and could be sharpened.
  • The method suggests a route to explicit countable parafree groups with nonzero H2: if a group homomorphism from a free group to a lamplighter-type group can be arranged so that a non-rational series arises from a cycle in the group's exterior square, the existence proof via HZ-localization could be replaced by concrete relations.
  • A natural testable extension is whether H2(\hat f, Z) contains a copy of Z[1/2] rather than just one 2-divisible element; the paper does not address the full structure of this homology group.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies homological properties of parafree Lie algebras. In Theorem A it constructs an explicit countable parafree Lie algebra over a field of characteristic 2 with nonzero second homology. In Theorem B it claims that the pronilpotent completion of the free Lie algebra of rank two over Z has cohomological dimension at least 3, by constructing a nonzero 2-divisible element in H2. Section 2.4 proves existence of countable parafree groups with nonzero H2, following Bousfield's theorem. The proofs use a Lie-algebra analog of Stallings' theorem, a Lie analog of the lamplighter group, and non-rationality of lacunary power series.

Significance. If the proofs are completed, Theorem A would give the first explicit example of a parafree Lie algebra with nontrivial second homology, and Theorem B would settle a Lie-algebra counterpart of a question about cohomological dimension of pronilpotent completions. The group-theoretic result in Section 2.4 is also new. The overall approach is original, and the idea of detecting nonzero homology via projections to a lamplighter Lie algebra and non-rational power series is elegant. The paper is clearly structured and mostly self-contained, with useful references to prior work.

major comments (2)
  1. [Section 2.3, definition of s_{2^n}] The displayed definition of s_{2^n} in the proof of Theorem B has upper summation limit 2^n-1, whereas the identity quoted from [13, Lemma 4.1] and the definition used in Theorem 2.11 require upper limit n. With the printed upper limit the element t_{2^n} is not a cycle: for n=2, a direct Jacobi expansion gives [r_4,a]+[s_4,b] = [[a,3b],[a,b]] ≠ 0 in the free Lie algebra f_Z(a,b). Therefore the infinite sum ∑ 2^n t_{2^n} cannot be claimed to represent an element of H2(ĥf,Z) via Proposition 2.2. This is a load-bearing error for Theorem B and must be corrected, presumably by setting the upper limit to n, and the cycle condition re-verified.
  2. [Section 2.1, Theorem 2.11 and Section 2.3, Theorem B] The proofs of Theorem 2.11 and Theorem B treat infinite sums such as R_α = ∑ α_n r_{2^n}, S_α = ∑ α_n s_{2^n}, and ∑ 2^n t_{2^n} as elements of the exterior square ĥf ∧ ĥf and then as cycles in Ker(ĥf ∧ ĥf → ĥf). However, Proposition 2.2 is stated for the algebraic exterior square of a discrete Lie algebra, and the algebraic exterior square does not contain arbitrary infinite sums. The paper neither defines a completed exterior square nor proves the natural isomorphism H2(ĥg) ≅ Ker(ĥg ∧̂ ĥg → ĥg) for pronilpotent completions. Without such a result, the nonzero-homology conclusions in Theorem 2.11 and Theorem B do not follow from the written arguments. The authors should either define the completed construction and establish the isomorphism, or provide an alternative argument via finite truncations and inverse limits.
minor comments (3)
  1. [Notation throughout] The expression '2^{n-i}b' is printed as '2n−ib' in several places, which is ambiguous; it should be typeset as a power of 2.
  2. [Section 2.3, 2-divisibility argument] The equality ∑ 2^n t_{2^n} = 2^k ∑_{n≥k+1} 2^{n-k} t_{2^n} in H2(ĥf,Z) relies on the same unproved completed-exterior-square identification; after the main gap is fixed, the authors should also justify that the infinite tail again defines a homology class.
  3. [Section 2.2, proof of Lemma 2.13] The argument that r ∩ [e,e] = [r,e] in the proof of Lemma 2.13 is compact; a sentence explaining why the K-linear span of the α_i, β_i contributes nothing to r ∩ [e,e] would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constructions are explicit and self-contained, with only a non-load-bearing self-citation for an algebraic identity.

full rationale

The paper's central claims are derived by explicit construction against external benchmarks. Theorem A defines the Lie algebra b directly by generators and relations, proves H2(b)=0 via Hopf's formula, obtains parafreeness by applying Stallings' theorem for Lie algebras, and proves H2(a) is nonzero by projecting onto the lamplighter Lie algebra lK and detecting nonzero classes through non-rationality of formal power series. Theorem B similarly constructs an explicit cycle in the exterior square of a free Lie algebra and proves 2-divisibility of the resulting homology class by direct manipulation of the defining sums. The only citation that carries real weight is the identity quoted from [13, Lemma 4.1] and used to verify the cycle condition [r_{2^n},a]+[s_{2^n},b]=0. That citation is to the authors' earlier work, but it is a parameter-free algebraic identity whose statement does not include the paper's target results, so it is not equivalent to the conclusions by construction. There are no fitted inputs called predictions, no renaming of known results, and no imported uniqueness theorem. The proof does contain a potentially serious correctness gap concerning infinite sums in completed exterior squares, but that is a rigor issue, not circularity. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

The central claims rely on standard homological algebra, on the authors' published identity from their earlier paper, and on deep results about HZ-localization and Gamma-systems. No parameters are fitted; no new entities are postulated beyond the explicitly defined lamplighter Lie algebra, which is a tool rather than an unexplained assumption.

assumptions (7)
  • domain assumption All Lie algebras and homologies are taken over a commutative associative ring R, with K a field of characteristic two for Theorem A.
    This is the setting stated in Section 1; Theorem A requires characteristic two for the Engel identities in Lemma 2.12.
  • standard math Hopf's formula and the Ellis exterior-square description H2(g,R) is isomorphic to Ker(g wedge g to g).
    Used throughout Sections 2.1 to 2.3 to represent second homology classes by elements of the exterior square.
  • standard math Stallings' theorem for Lie algebras: a 2-connected map induces isomorphisms on lower central quotients.
    Proved in the paper as Theorem 2.14 by a five-term exact sequence comparison; this is the engine for parafreeness in Lemma 2.15.
  • standard math The Engel identity quoted from the authors' earlier paper is valid in any Lie algebra.
    Used to show [R_alpha,a] + [S_alpha,b] = 0, i.e. that the proposed elements are cycles; it is a self-cited published lemma from [13, Lemma 4.1].
  • domain assumption Bousfield's theorem that H2 of the pronilpotent completion of a noncyclic free group is uncountable.
    Used in Section 2.4 together with filtered colimits to conclude that some countable parafree subgroup has nonzero H2.
  • domain assumption Farjoun-Orr-Shelah results on HZ-localization, Gamma-systems of equations, and the equality of Gamma-closure with HZ-localization, with countable systems sufficient.
    Load-bearing for Proposition 2.19 and Corollary 2.20, which produce countable parafree subgroups inside the pronilpotent completion.
  • domain assumption The exterior square and homology of the pronilpotent completion are compatible with the inverse-limit topology, so infinite sums such as sum 2^n t_{2^n} are legitimate cycles.
    Used without explicit definition in Sections 2.1 and 2.3; required for Theorem 2.11 and Theorem B.
invented entities (1)
  • The lamplighter Lie algebra l_R = R[x] semidirect R t with action [p,t] = p x independent evidence
    purpose: Detection object used to project homology classes from the free Lie algebra completion and to show nonvanishing via non-rational formal power series
    Introduced explicitly in Definition 2.4 as a semidirect product; it is a constructive tool rather than a postulated unexplained entity, and its structure is fully specified.

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Pith. "Pith review of Homological properties of parafree Lie algebras." pith.science (2026). https://pith.science/paper/ITZMPH4X

@misc{pith2026190804608,
  author       = {Pith},
  title        = {Pith review of: Homological properties of parafree Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ITZMPH4X}},
  note         = {Machine review of arXiv:1908.04608}
}
abstract

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.

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