{"id":"635f997a-1850-4802-8639-ecf880bf060a","arxiv_id":"2509.01559","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Steinberg homology vanishes in a range for all reductive groups, and the double Tits building T^2(Z^n) is n-connected, refining the Church-Farb-Putman conjecture in degrees 1 and 2.","lead":"This paper proves that the homology of any connected reductive group over a field, with coefficients in the Steinberg representation, vanishes in a range of degrees depending on the group's root system. It also shows that a certain simplicial complex associated to the integer matrices SL_n(Z), the double Tits building, is highly connected, giving new evidence for a conjecture about the cohomology of SL_n(Z).","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem D hinges on exactness of the X2→X1→X0→St resolution, which is delegated to [9] without verifying that the enlarged X2 preserves exactness at X1.","rationale":"The reader's weakest-assumption analysis points to Theorem 31.5, and my reading agrees: this is the most load-bearing unsupported step. The paper's main field-theoretic theorem (Theorem B) has a long but internally coherent spectral sequence proof; I did not find a concrete flaw there. The integral part, however, rests on two external citations: Theorem 23.3 from [22] and Theorem 31.5 from [9]. Of these, Theorem 31.5 is the more delicate because the paper modifies X2 while claiming exactness is unaffected. The concern is not that the authors are wrong; it is that the proof does not establish the exactness at the level of detail needed for a result that is otherwise new and substantial. A second, more minor issue is the misprint in the general-ring bound in the statement of Theorem C in the introduction: it reads ⌊(n−3)/3⌋, while the proof and Corollary E require ⌊(n−2)/3⌋. This is a typo, not a mathematical gap, and I would not base a rejection on it. The verdict should remain conditional: the paper is promising and likely correct, but the exactness of the three-step X• resolution should be either proved in full or pinned to an exact theorem in [9] before the connectivity theorem is relied upon.","tokens_in":75511,"tokens_out":20527,"duration_ms":228988,"concrete_test":"Independently verify Theorem 31.5 for small ranks. Take V=Z^3 and V=Z^4, write down X2, X1, X0 explicitly from the definitions, and compute the homology H1(X•(V)) = ker(X1→X0)/im(X2→X1) by direct linear algebra (or a short Sage/Magma script). If H1 is nonzero for either rank, Theorem D collapses. Also extract from [9] the exact statement and differential of their X2-term and check that this paper's X2 contains it as a subcomplex with the same restriction of δ. If the containment and δ-commutation are not literal, the proof of Theorem 31.5 is incomplete and the exactness claim must be proved independently.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's new connectivity theorem for T^2(Z^n) (Theorem D) and the resulting Corollary E depend on Theorem 31.5: the truncated resolution X2(V)→X1(V)→X0(V)→St(V)→0 is exact. This exactness is load-bearing because Lemma 32.2 uses precisely H0 and H1 of X•(V) to identify the homology of the double complex with that of S•(Z^n); if H1(X•(V)) is nonzero, the comparison fails and Theorem D does not follow. The proof of Theorem 31.5 is a reference-and-comparison argument: it says that X0 and X1 agree with the corresponding terms in [9], and that X2 is larger, so no claim about the kernel of X2→X1 is affected. That argument is valid only if the enlarged X2 is a genuine chain complex over the same X1 and contains the [9] X2-term as a subcomplex whose image already covers ker(X1→X0). The paper does not display the [9] differential or verify that the present larger X2 restricts to it; it only points to Examples 31.2–31.4. If [9]'s X1 module carries extra relations not present here, or if the added X2 generators do not map into the required kernel, exactness at X0 or at X1 could fail. The authors' terse 'this does not affect the result' is not a proof, and the result is central rather than peripheral.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for any connected reductive group G over a field k, the homology of G(k) with coefficients in the Steinberg representation vanishes in a range depending only on the relative root system Φ_k(G) (Theorem B). For the classical split groups this was known by Ash–Putman–Sam; the new proof is uniform and extends to all reductive groups, including nonsplit forms and the nonreduced system BC_n. The paper also states an integral refinement: assuming a high-connectivity conjecture for the double Tits building T^2(Z^n), it obtains vanishing for H_i(SL_n(Z); St) and H_i(GL_n(Z); St) in a range that improves with the connectivity assumption (Theorem C). The main new unconditional evidence is Theorem D, asserting that T^2(Z^n) is n-connected for n ≥ 4, which yields, via Theorem C, new low-degree vanishing for SL_n(Z) and GL_n(Z) (Corollary E). The proofs use a resolution of the Steinberg representation, a spectral sequence built from Levi subgroups, and, for the integral part, a double complex of partial resolutions.","tokens_in":75812,"tokens_out":13839,"duration_ms":141627,"significance":"If the gaps identified below are repaired, this is a substantial contribution. Theorem B unifies and extends prior vanishing results to all reductive groups over fields, with explicit bounds depending on the relative root system; the spectral sequence machinery and the explicit differential calculations in Parts 2–4 are detailed and appear sound. The integral part gives a clean conditional framework for the Church–Farb–Putman conjecture and proves a genuine new connectivity result for the double Tits building, leading to new low-degree vanishing for SL_n(Z) and GL_n(Z). The paper is careful to attribute prior work and states the precise dependency on external results (Reeder’s theorem, Solomon–Tits, and Miller–Patzt–Wilson’s identification of the bar resolution with the double Tits building). These strengths are real, but the load-bearing omissions discussed in the major comments must be addressed before the result can be considered fully verified.","major_comments":[{"comment":"The exactness of X2(V)→X1(V)→X0(V)→St(V)→0 is asserted by comparison with [9], with the comment that enlarging X2 does not affect the claim because no assertion is made about ker(X2→X1). This is load-bearing: Lemma 32.2 requires H_i(X•(V))=0 for i=0,1 for every summand, and Theorem D′ (hence Theorem D and Corollary E) would fail if H_1(X•(V)) were nonzero. The argument is valid only if the enlarged X2(V) is a genuine chain complex with δ²=0 and its image lies in ker(X1→X0), and if the image of the [9] X2-term already covers that kernel. None of these is demonstrated; the text refers only to Examples 31.2–31.4. Please provide a direct verification for the new generators (especially the multi-line forms in Example 31.4) or quote and verify the precise statement from [9].","section":"§31.3, Theorem 31.5; used in §32.2, Lemma 32.2"},{"comment":"The proof of Theorem C′.1 is omitted with the note that it follows Part 2 closely after replacing the spectral sequence and the reducible-Levi results. This theorem supplies the 2-and-3-invertible cases of Corollary E, one of the paper's main concrete applications. The listed replacements are plausible, but the key differential lemmas (the analogues of Lemmas 12.1–12.3) are not stated or checked, and the induction with the cap min(b,·) needs verification. I recommend writing out the full proof or at least an appendix containing the integral analogues of the differential lemmas with precise statements.","section":"§25.4 (Theorem C′.1)"},{"comment":"The statement of Theorem C gives the general-coefficient vanishing range as i ≤ min(b, ⌊(n−3)/3⌋). This is inconsistent with Theorem C′ (§25.2), which for GL_{n+1} gives i ≤ min(b, ⌊(n−1)/3⌋), i.e., for GL_n, i ≤ min(b, ⌊(n−2)/3⌋). Corollary E's thresholds n≥5 for i=1 and n≥8 for i=2 match the latter. The formula in Theorem C should be corrected to ⌊(n−2)/3⌋ (or the statement of Theorem C′ should be aligned if the intended bound is different).","section":"§1.9, Theorem C"}],"minor_comments":[{"comment":"Typo: “The generalizes work” should be “This generalizes work”; “first and second of homology” should be “first and second homology groups.”","section":"Abstract"},{"comment":"The table and surrounding text are accurate but the sentence “In fact, with only a little more effort the proofs in [9, 15] work for F a commutative ring in which all primes p ≤ n are invertible” is slightly imprecise: the statement is correct for the specific claims used, but the reader must infer the exact scope. Consider making the coefficient condition explicit in the table.","section":"§1.9, Remark 1.18"},{"comment":"The boundary formula for the bar complex is written with signs (−1)^{j−1}; this is fine, but the signs in the double-complex differential δ in §32.2 (with the shifts by r_1+...+r_{j−1}+i_1+...+i_{j−1}) deserve a small justification, especially because Lemma 32.2 relies on the signed differentials being compatible with the tensor-product differentials. Currently the verification is only hinted at.","section":"§30.4, formula for ∂"},{"comment":"The paper is very long and some parts are repetitive (e.g., the three parallel treatments of types A, B/C/BC, and D). This is a presentation issue rather than a correctness issue.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are likely correct, but the manuscript in its current form has three issues that need attention before acceptance: the terse verification of Theorem 31.5 (which is load-bearing for the double-Tits-building result), the omitted proof of Theorem C′.1, and an off-by-one error in the statement of Theorem C. The first two are omissions of proof rather than errors, and the third is a typo, so I recommend major revision rather than rejection. I would also encourage the editor to ask the authors to clarify the relationship with the forthcoming companion paper [21], since Theorem 23.3 is extracted from [22] and a promise of a different proof in [21] is made but not used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a serious paper with a real result. Theorem B extends Ash–Putman–Sam's Steinberg vanishing to all connected reductive groups over arbitrary fields, and the machinery—relative root systems plus a spectral sequence built from Reeder's map—is a genuine improvement over the original. Theorem D, the n-connectivity of the double Tits building T2(Zn), is new and, if it holds, is the first non-tautological evidence for the Church–Farb–Putman conjecture. I traced the main lines of Parts 1–4; the spectral sequence arguments are detailed and the differential calculations are explicit. The reductions to irreducible types are careful, and I didn't find a fatal flaw in Theorem B.\n\nBut there are two spots where the paper asks you to take a lot on faith, and both are load-bearing. First, Theorem 31.5—the exactness of X2→X1→X0→St—is the foundation for Theorem D. The proof is a reference to [9] plus the claim that enlarging X2 \"does not affect the result.\" That is not a proof of exactness at X1. A larger X2 can only enlarge the image of the differential, so you still need to check that the new generators map into the kernel of X1→X0; otherwise you don't have a chain complex. The paper doesn't display the differentials or verify the chain condition. This is probably fixable, but it's not a formality.\n\nSecond, Theorem C′.1—the case where 2 and 3 are invertible—is stated with a vanishing range, but the proof is essentially \"follows Part 2 closely with these changes.\" For a central theorem, that's too thin. A referee needs to see the analogue of the type-A surjectivity arguments, or at least a precise list of which lemmas carry over.\n\nMinor but real: the statement of Theorem C in the introduction has misprinted bounds (⌊(n−2)/2⌋ and ⌊(n−3)/3⌋); the correct bounds appear in Theorem C′ (⌊(n−1)/2⌋ and ⌊(n−1)/3⌋), and Corollary E uses the correct ones. The intro needs cleaning up.\n\nThe integral part leans heavily on [9] and [22], both by overlapping authors, but I don't see circularity in Theorem B—the new vanishing theorem doesn't depend on Theorem D.\n\nWho is this for? People working on the cohomology of arithmetic groups and Steinberg modules. It deserves a serious referee, but the referee should be asked to focus on Theorem 31.5 and the C′.1 reduction. My recommendation: send to peer review; require a real verification of Theorem 31.5 and a more complete argument for C′.1; fix the Theorem C statement.","headline":"Serious, substantial paper with Theorem B extending Steinberg vanishing to all reductive groups; Theorem D is new but rests on a deferred exactness proof and a thinly sketched case, so it needs revision, not desk rejection.","tokens_in":76330,"tokens_out":5486,"would_cite":true,"duration_ms":56580,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20J06","20G10","11F75","51E24"],"pacs":[],"model":"deepseek-v4-flash","headline":"Steinberg homology vanishes in a range for every reductive group.","keywords":["Steinberg representation","reductive groups","homological vanishing","Tits building","relative root systems","arithmetic groups","double Tits building","SL_n(Z) cohomology"],"falsifier":"Search the finite generating sets of Examples 31.3 and 31.4 for rank 4 or 5: if any element of X_1(V) is killed by the boundary to X_0(V) but is not in the image of the boundary from X_2(V), then Theorem 31.5 and hence Theorem D are false. Equivalently, a direct computation of H_1 of the bar resolution S_bullet(Z^4) that produced a nonzero class would settle the question in the negative.","tokens_in":75373,"feed_emoji":"","tokens_out":8269,"duration_ms":93522,"temperature":0.7,"pith_summary":"The paper's central claim is that for any connected reductive group G over a field k, the group homology of G(k) with coefficients in the Steinberg representation vanishes up to a degree depending only on the relative root system of G, and this holds over every commutative coefficient ring. Previous arguments covered the split classical groups; here the proof is reorganized around a resolution whose terms are built from the Steinberg representations of Levi subgroups, plus a spectral sequence whose differentials can be computed. The same construction is carried to the integers: the paper states a connectivity conjecture about the double Tits building of pairs of compatible flags of direct summands of Z^n, and shows this connectivity would yield the long-sought vanishing in the stable range for SL_n(Z). It proves the required connectivity for the first nontrivial case, showing the double building is n-connected for n at least 4, and thereby refines what is known in low degrees for GL_n(Z) and SL_n(Z) with Steinberg coefficients.","feed_headline":"Steinberg homology vanishes in a range for every reductive group","feed_subtitle":"Extends the split-case result to all reductive groups and sharpens low-degree integral cases.","key_machinery":"The engine is a resolution of St(G) whose degree-i term is a sum, over standard Levi subgroups obtained by deleting i+1 simple roots, of induced Steinberg representations of those Levi subgroups. Feeding this resolution into group homology gives a spectral sequence whose E^1 page is built from smaller-rank groups; the induction is driven by the parabolic induction formula for Steinberg representations and by vanishing and surjectivity statements for the end Levi factors. For the integer theorems the load-bearing object is the double Tits building T^2(Z^n), the simplicial complex of pairs of compatible flags of direct summands; a prior theorem identifies its homology with that of the bar reso","core_discovery":"On the paper's own terms, the discovery is that homological vanishing for Steinberg representations is a formal consequence of the structure of relative root systems, not of special features of split groups. Theorem B states that H_i(G(k); St(G;F)) = 0 for i at most b(Phi_k(G)), with b(A_n) = floor((n-1)/2), b(B_n) = b(C_n) = b(BC_n) = floor((n-2)/2), b(D_n) = floor((n-3)/2), b = 0 for exceptional systems, and for reducible systems a sum of the component bounds plus m-1. In the integral setting, Theorem D states that the double Tits building T^2(Z^n) is n-connected for n at least 4, and combined with a conditional spectral sequence it yields Corollary E: H_1 and H_2 with Steinberg coefficien","pith_inferences":["My inference: the relative-root-system formulation suggests analogous integral vanishings for other reductive Z-forms, such as symplectic or orthogonal groups, should follow from a connectivity conjecture for the appropriate integral double buildings; the paper explicitly says the approach can be generalized but does not write the details.","My inference: the 1/3 slope in the integral theorem appears forced by the base case SL_3(Z), where surjectivity is only verified after inverting 2 and 3; a sharper base case or a different resolution might make a 1/2 slope accessible, but the paper's own remark indicates its spectral sequence alone cannot achieve that.","My inference: the combinatorial restatement in Lemma 33.1 turns each summand of the connectivity proof into a finite statement about partitions of a set respecting specified subsets, so low-rank cases could be checked by computer search and might indicate whether higher cases of the b-integral resolution conjecture hold.","My inference: a natural testable extension is to replace Z by Z[1/N] or rings of integers in number fields; the bar-resolution formalism and the double-building definition carry over, and the first obstruction would be the analogue of the three-step partial resolution for those coefficient rings."],"forward_implications":["Theorem B gives a vanishing range for nonsplit reductive groups, so Steinberg-cohomology vanishing now applies to every connected reductive group over any field.","The proof replaces topology of partial flag complexes with a self-contained spectral sequence from a Levi-subgroup resolution, and the same machinery drives the integral results.","For exceptional relative root systems the theorem yields only H_0 vanishing, but the paper notes that a less uniform argument would raise the bounds for E_6, E_7, and E_8.","The n-connectivity of T^2(Z^n) for n at least 4 yields new integral vanishing: H_1 for SL_n(Z) and GL_n(Z) in ranges starting near n=4 or 5, and H_2 near n=6 or 8, depending on whether 2 and 3 are invertible in the coefficient ring.","If the connectivity assumption in Theorem C holds for larger b, the same mechanism proves the conjectured stable-range vanishing for SL_n(Z) and GL_n(Z) up to degree b for large n."],"supporting_citations":[{"why":"Supplies the prior vanishing theorem for the classical split groups that the paper extends to all reductive groups.","marker":"[2]"},{"why":"Provides the three-step partial resolution of the integral Steinberg module from modular-symbol generators used in the proof of Theorem D.","marker":"[9]"},{"why":"States the stability conjecture for high-dimensional cohomology of SL_n(Z) that motivates the integral vanishing results.","marker":"[14]"},{"why":"Gives the presentation of the Steinberg module and the degree-1 vanishing used as base cases for the integral theorem.","marker":"[15]"},{"why":"Proves the degree-0 vanishing over SL_n(Z) used in the integral base cases.","marker":"[18]"},{"why":"Provides the bar-resolution form of the rational resolution and related exactness results used to set up the spectral sequence.","marker":"[19]"},{"why":"Relates homology of the bar resolution to connectivity of the double Tits building, the bridge for the integral half.","marker":"[22]"},{"why":"Gives the parabolic induction formula for Steinberg representations that underlies the resolution and spectral sequence for reductive groups.","marker":"[25]"},{"why":"Identifies the Tits building as a wedge of spheres and defines the Steinberg representation as its top reduced homology.","marker":"[28]"}],"fun_headline_variants":["Steinberg homology vanishes for all reductive groups","Reductive groups: Steinberg cohomology vanishing extended","Vanishing theorem for Steinberg homology, now in full generality","Steinberg vanishing: new proof for any reductive group","Integral Steinberg homology: vanishing range and refinements"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"For the integral half, the load-bearing premise is the exactness of the truncated three-step resolution X_2(V) -> X_1(V) -> X_0(V) -> St(V) -> 0; the paper verifies it by saying the generators and relations match a cited source and that enlarging X_2 does not affect the claim, rather than giving a full detailed proof, and if this exactness fails the connectivity of T^2(Z^n) would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Steinberg homology vanishes for all reductive groups","Reductive groups: Steinberg cohomology vanishing extended","Vanishing theorem for Steinberg homology, now in full generality","Steinberg vanishing: new proof for any reductive group","Integral Steinberg homology: vanishing range and refinements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000175,"raw_usage":{"total_tokens":1088,"prompt_tokens":672,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":348}},"tokens_in":416,"tokens_out":416,"duration_ms":4817,"temperature":1.0,"reasoning_tokens":348,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:25:38.222629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the finite generating sets of Examples 31.3 and 31.4 for rank 4 or 5: if any element of X_1(V) is killed by the boundary to X_0(V) but is not in the image of the boundary from X_2(V), then Theorem 31.5 and hence Theorem D are false. Equivalently, a direct computation of H_1 of the bar resolution S_bullet(Z^4) that produced a nonzero class would settle the question in the negative.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prior vanishing theorem for the classical split groups that the paper extends to all reductive groups."},{"cited_title":"Br¨ uck, J","cited_arxiv_id":null,"evidence_quote":"Provides the three-step partial resolution of the integral Steinberg module from modular-symbol generators used in the proof of Theorem D."},{"cited_title":"Algebraic Topology: Applications and New Directions","cited_arxiv_id":null,"evidence_quote":"States the stability conjecture for high-dimensional cohomology of SL_n(Z) that motivates the integral vanishing results."},{"cited_title":"Church & A","cited_arxiv_id":null,"evidence_quote":"Gives the presentation of the Steinberg module and the degree-1 vanishing used as base cases for the integral theorem."},{"cited_title":"Lee & R.H","cited_arxiv_id":null,"evidence_quote":"Proves the degree-0 vanishing over SL_n(Z) used in the integral base cases."},{"cited_title":"Miller, R","cited_arxiv_id":null,"evidence_quote":"Provides the bar-resolution form of the rational resolution and related exactness results used to set up the spectral sequence."},{"cited_title":"Reeder, The Steinberg module and the cohomology of arithmetic groups, J","cited_arxiv_id":null,"evidence_quote":"Gives the parabolic induction formula for Steinberg representations that underlies the resolution and spectral sequence for reductive groups."},{"cited_title":"Solomon, The Steinberg character of a finite group with BN -pair, in Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), 213–221, Benjamin, New York","cited_arxiv_id":null,"evidence_quote":"Identifies the Tits building as a wedge of spheres and defines the Steinberg representation as its top reduced homology."}],"review_version":1}