{"id":"521d2c72-7cef-4ac8-8868-9c10083ed602","arxiv_id":"2411.13272","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Homological Sigma-invariants are defined for locally compact Hausdorff groups and shown to recover and extend the classical discrete-group invariants.","lead":"This paper defines homological Sigma-invariants for locally compact Hausdorff groups, extending a classic tool from discrete group theory to groups that carry a topology. It proves criteria for finiteness properties of such groups and recovers the classical discrete theory as a special case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8.7 composes 'finitely modeled' chain endomorphisms and inclusions without proving the class is closed under composition; if a composition fails the shape or G-finite bullets, the main equivalence (Theorem F/G) does not follow.","rationale":"The reader's weakest assumption flagged the 'finitely modeled' notion as fragile; this stress-test identifies a concrete, load-bearing manifestation of that fragility: the proof of Theorem 8.7 repeatedly composes finitely modeled maps without proving composition closure. The central claims (Theorem F/G, Theorem 8.7) depend on those composites being finitely modeled, so the gap is directly tied to the paper's main equivalence. The concern is likely fixable—closure under composition probably holds via the length bounds of Lemma 8.3 and finite shape counts—but it must be proven. Since the reader's verdict was already CONDITIONAL, my read does not move the verdict; it refines the condition to include a closure lemma for the finitely modeled class.","tokens_in":39340,"tokens_out":20208,"duration_ms":197227,"concrete_test":"Prove or disprove closure under composition: for finitely modeled ε:C_*(VR_k)→Z[G] and δ:C_*(VR_l)→Z[G] with im ε ⊆ C_*(VR_l), verify all four bullets of Section 8 for δ∘ε. If a counterexample exists, apply it to the specific maps in Theorem 8.7 (the inclusion VR_k→VR_n and φ^{∘i+1}∘μ). A direct check: for G=Z with χ=id and m=1, compute the composite ι∘φ∘μ explicitly, verify the 1-shape condition and the G-finite condition, and see whether any bullet fails; a failure would invalidate Theorem 8.7.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 8 defines a chain endomorphism to be 'finitely modeled' via four bullets: shape conditions for q=1, connected nondegenerate centric shapes for q≥2, and a G-finite condition on ε(τ)(0)\\ε∂(τ)(0). No lemma states that this class is closed under composition or under pre/post-composition with inclusions VR_k→VR_n. Yet the proof of Theorem 8.7 (and hence Theorem G) relies on such compositions: in condition 1→2 it forms ι_{lk}∘φ_{k,*} or μ∘φ^{∘i+1}, and in condition 2→1 it forms φ^{∘i+1}_{*}∘μ_{*}, with source VR_n and target VR_k. These composites are asserted to be the finitely modeled witnesses required by Theorem 8.6(3). The shape conditions are not obviously preserved: ε may map a simplex to a sum of many simplices, and applying δ to each summand can produce chains whose total shape is not among finitely many connected, centric, nondegenerate shapes, nor is the G-finite difference condition automatic. Without a closure proof, the constructed maps may fail the definition, leaving the central criteria of the paper (Theorems 8.6, 8.7, F, G) with an unverified step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a homological Sigma-invariant for locally compact Hausdorff groups, denoted Sigma^m_top(G;Z), using the filtration of the standard free resolution by Vietoris-Rips complexes. It proves that 0 belongs to the invariant exactly when G is of type CP_m, establishes a Hurewicz-type comparison with the homotopical invariant from the companion paper [BHQ24], recovers the classical Sigma^m(G;Z) for discrete groups of type FP_m, and gives criteria for membership in terms of 'finitely modeled' chain endomorphisms that raise a character valuation. It then proves openness of the invariant and several extension theorems for short exact sequences, including the abelian-quotient criterion for kernels of type CP_m.","tokens_in":924,"tokens_out":1603,"duration_ms":111071,"significance":"These are substantial new results: if correct, they establish the expected homological Sigma-theory for locally compact groups, with Theorem D tying the new invariant to classical Bieri-Neumann-Strebel-Renz theory and Theorems F and G giving concrete chain-endomorphism criteria. The paper is not programmatic: it contains detailed constructive proofs, especially the compactness argument that produces finitely modeled chain maps in Theorems 8.4 and 8.6, and it is explicit about the companion-paper dependency. The main risk is that the newly introduced 'finitely modeled' condition is used under compositions in the central criteria, yet no closure lemma for that class is proved. The overall framework and theorems are plausible, but this missing step is load-bearing.","major_comments":[{"comment":"The proof of Theorem 8.7 composes finitely modeled chain endomorphisms in several places, e.g. iota_lk composed with phi_{k,*}, mu_{l,*} composed with phi_l^{circ i+1} composed with iota_kl, and phi_*^{circ i+1} composed with mu_*, and asserts without proof that the resulting maps are again finitely modeled. No lemma states that the class of finitely modeled chain maps is closed under composition, pre-composition with inclusions, or post-composition with inclusions. This is not automatic: for q=1 the definition requires the output on a basis simplex to have one of finitely many connected shapes; for q>=2 the definition requires a decomposition compatible with the connected components of epsilon_{q-1} partial_q(sigma), and under composition those components can split or merge. The G-finite difference condition is likewise not shown to be stable. Since Theorems 8.6, 8.7, F, and G rely on these composites as witnesses, this is an unverified load-bearing step. The authors should add a closure lemma (or modify the definition/construction so that closure is evident) before the main equivalences can be accepted.","section":"Section 8, definition of 'finitely modeled'; Theorem 8.7"},{"comment":"The proof of condition 3 implies condition 1 fixes a single chain endomorphism phi on C_*(VR_{eK}(G)), but then applies phi to chains in VR_{l1}(G) and uses a chain homotopy on C_{m-1}(VR_k(G)); the text says 'we assume that phi_{l2}, phi_{l1}, phi_k match when restricted and denote all of them by phi.' No argument is given that the family of endomorphisms supplied by condition 3 is compatible under restriction or that the index eK can be chosen after l1 and l2 are chosen. Since condition 3 provides a map for every sufficiently large index, this may be repairable, but as written the index choices and restriction procedure are not demonstrated, and this direction is needed for Theorem F.","section":"Theorem 8.6, implication 3 implies 1"}],"minor_comments":[{"comment":"The statement of Theorem G says 'there exists k >= 0 such that' but the proof and the surrounding text indicate that the intended meaning is 'there exists k such that for all sufficiently large indices the equivalence holds', or at least that k is the index associated to the homological connecting vector. Please clarify the quantifier order.","section":"Section 1, Theorem G"},{"comment":"The definition of Sigma^1_top(G;R) uses 'for some c >= 0' but does not specify that c must be large enough or that the homology group is computed for the chain complex R[Delta^q_c cap (G_chi)^{q+1}]. This is a minor clarity issue, as the intended meaning is clear from context.","section":"Section 5, Definition 5.1"},{"comment":"In the proof of Lemma 8.1(3), the induction step replaces vertices x0 and x1 by new vertices y0 and y1. It should be explicitly stated that the new vertices are chosen outside the union of the supports of S and its boundary so that the replacement does not create accidental identifications.","section":"Section 8, Lemma 8.1"},{"comment":"In the proof of part 1, the construction of the chain homotopy lambda uses the phrase 'Without loss of generality m_{q-1} >= n_q'. The reason this lossless assumption is valid should be spelled out, since m_{q-1} and n_q are constructed separately.","section":"Section 10, Theorem 10.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and largely well-written, but the missing closure property for 'finitely modeled' chain maps is a genuine gap in the proof of the central Theorem 8.7 (and thus Theorems F and G). The authors should either prove the closure or adjust the definition and constructions so that the composites used in the proof are visibly finitely modeled. This is a fixable issue within the scope of the manuscript, hence major_revision rather than reject. There is also a minor quantifier ambiguity in Theorem G that should be corrected during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is the real deal: a homological Sigma-invariant for locally compact Hausdorff groups, with a Hurewicz-type comparison to the homotopical version, recovery of the classical Bieri–Renz Sigma invariant in the discrete case, and criteria for type CP_m via chain endomorphisms on Vietoris–Rips complexes. If the machinery works, it is the right generalization and the natural home for future Sigma-theory computations in the locally compact setting. Second, there is a specific spot I would want a referee to attack: Theorem 8.7 composes 'finitely modeled' chain endomorphisms and asserts the composites are finitely modeled, but the paper never states or proves closure of that class under composition. It is not obvious from the definition. The four bullets (shape conditions plus G-finite difference) control individual basis simplices, not linear combinations of their images, so a sum of two shape-controlled chains can have a connected component whose shape is new. The proof of 8.7 needs a closure lemma or a different construction. I do not think this is fatal—probably a missing lemma—but it is exactly the kind of thing that can break the main equivalence.\n\nThe good parts: the paper is well organized, the main theorems are stated crisply, and the proofs are detailed rather than hand-wavy. The extension theorems (J and K) look useful, and the openness result (Theorem H) is a nice touch. The paper depends heavily on the companion homotopical paper [BHQ24], importing several key lemmas as black boxes; that is normal for a two-paper project but means the two should be refereed together.\n\nVerdict: I would accept this for peer review and ask for revision. A good referee should check the composition issue and whether the 'finitely modeled' notion can be replaced by something manifestly closed under composition. For a reader in geometric group theory, this is worth reading once the companion paper is out. I would probably cite it for the definition even if the criteria need more work.","headline":"A serious and mostly careful homological Sigma-theory for locally compact groups, but the composition of finitely modeled maps in Theorem 8.7 needs a missing closure lemma before the main criteria are fully solid.","tokens_in":40133,"tokens_out":4121,"would_cite":true,"duration_ms":46327,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20J05","20F65","22D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines a new homological Sigma-invariant for locally compact Hausdorff groups that recovers the classical Sigma-invariant for discrete groups and characterizes membership through finitely modeled chain endomorphisms.","keywords":["Sigma-invariants","locally compact groups","type CP_m","homological finiteness","Vietoris-Rips complex","valuations","chain endomorphisms","group extensions"],"falsifier":"Compute $\\Sigma^2_{\\mathrm{top}}(G;\\mathbb Z)$ for the discrete group $G=\\mathrm{BS}(2,3)=\\langle a,b\\mid b^{-1}a^2b=a^3\\rangle$ directly from Definition 5.1 (where compact subsets are finite) and compare with the known classical $\\Sigma^2(G;\\mathbb Z)$; any disagreement would refute Theorem D, and the same computation tests whether the finitely modeled chain endomorphism criterion of Theorem F reproduces the same invariant.","tokens_in":39149,"feed_emoji":"📐","tokens_out":9808,"duration_ms":91445,"temperature":0.7,"pith_summary":"This paper develops the homological $\\Sigma$-theory for locally compact Hausdorff groups, defining invariants $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ that measure, in a given direction such as a character $\\chi\\colon G\\to\\mathbb R$, whether $G$ is of homological type $\\mathrm{CP}_m$ along that direction. The main theorems give concrete criteria: a nonzero character lies in $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ exactly when there are finitely modeled chain endomorphisms of the Vietoris–Rips chain complexes that raise the $\\chi$-valuation by arbitrarily large amounts, and a single such character detects type $\\mathrm{CP}_m$. For discrete groups of type $\\mathrm{FP}_m$, the new invariant agrees with the classical $\\Sigma^m(G;\\mathbb Z)$, so the construction genuinely extends the classical theory. A Hurewicz-type comparison with the homotopical locally compact invariant, transfer theorems for group extensions, and a criterion for closed normal subgroups with abelian quotient complete the picture.","feed_headline":"Homological finiteness invariant reaches locally compact groups","feed_subtitle":"It recovers the classical Sigma-invariant for discrete groups and gives chain-map criteria for membership.","key_machinery":"The machine carrying the argument is the Vietoris–Rips filtration $\\mathrm{VR}_k(G)$ of the free simplicial set on a locally compact group $G$, whose simplicial chain complexes $C_*(\\mathrm{VR}_k(G))$ filter the standard free resolution of $\\mathbb Z$ over $G$. For a character $\\chi$, the valuation $v$ on chains assigns to a simplex the minimum of $\\chi$ on its vertices, so the sublevel set $\\{v\\ge0\\}$ is exactly the chain complex of $\\mathrm{VR}_k(G_\\chi)$. The key technical notion is a finitely modeled chain endomorphism: a $\\mathbb ZG$-chain map extending the identity whose values on generating simplices are constrained by finitely many connected, nondegenerate (and in high dimensions, centric) shapes, with the image under the map being $G$-finite. This finiteness condition replaces the finite generation that is automatic in the discrete case and gives the theory its locally compact character.","core_discovery":"The central discovery is that the homological invariant $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ of a locally compact Hausdorff group is the correct directional refinement of type $\\mathrm{CP}_m$, and that it admits a purely algebraic characterization in terms of chain endomorphisms. Theorem F states that for a group of type $\\mathrm{CP}_m$ with a nonzero character $\\chi$, one has $\\chi\\in\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ if and only if for every sufficiently large $k$ there is a finitely modeled $\\mathbb ZG$-chain endomorphism of the $m$-skeleton of the Vietoris–Rips complex extending the identity on $\\mathbb Z$ and raising the valuation by at least $K$ for every prescribed $K$. Theorem D shows equality with the classical $\\Sigma^m(G;\\mathbb Z)$ for discrete groups of type $\\mathrm{FP}_m$. Theorem C relates the homological and homotopical invariants by $\\Sigma^m_{\\mathrm{top}}(G)=\\Sigma^2_{\\mathrm{top}}(G)\\cap\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ for $m\\ge2$ and by equality for $m=1$. Theorems J and K provide transfer results for short exact sequences $N\\to G\\to Q$ and for abelian quotients respectively.","pith_inferences":["If the finitely modeled condition is the correct finiteness notion, the analogous criterion should also exist for the homotopical invariant, unifying the two versions through the Hurewicz-type comparison.","The valuation-raising chain endomorphisms resemble the moving maps of classical Sigma-theory and may admit a coarse-geometric reformulation as coarse maps of $G_\\chi$ pushing points far in the $\\chi$-direction.","Theorems J and K could be iterated to compute $\\Sigma^m_{\\mathrm{top}}$ for solvable locally compact groups built from abelian extensions, paralleling the classical treatment of metabelian groups.","Because $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$ is a cone over an open set in $\\operatorname{Hom}_{\\mathrm{TopGr}}(G,\\mathbb R)$, the invariant is determined by an open subset of the character sphere, and the criteria in the paper are designed to make that subset computable."],"forward_implications":["For discrete groups of type $\\mathrm{FP}_m$, the equality $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)=\\Sigma^m(G;\\mathbb Z)$ means every existing computation of classical Sigma-invariants also computes the new invariant.","The chain-endomorphism criterion of Theorem F reduces the question of membership in $\\Sigma^m_{\\mathrm{top}}$ to constructing explicit maps that raise valuations, giving a practical route for concrete locally compact groups such as semidirect products.","The Hurewicz-type Theorem C shows that the homotopical invariant is determined by the homological one together with the second homotopical layer, for $m\\ge2$.","Theorem J supplies transfer tools for group extensions, letting one pull membership back from a quotient when the kernel is of type $\\mathrm{CP}_m$ and push it forward when the kernel is of type $\\mathrm{CP}_{m-1}$.","Theorem K provides a new criterion: a closed normal subgroup with abelian quotient is of type $\\mathrm{CP}_m$ whenever all characters vanishing on the subgroup already lie in $\\Sigma^m_{\\mathrm{top}}(G;\\mathbb Z)$."],"supporting_citations":[{"why":"Companion paper that develops the homotopical version of the invariants and supplies the lemmas (3.7, 3.5, Cor 5.4) that the homological proofs here import as black boxes.","marker":"[BHQ24]"},{"why":"Defines type C_m and CP_m for locally compact groups and provides Lemma 1.1.3 used in the Hurewicz argument.","marker":"[AT97]"},{"why":"Establishes the valuation-on-free-resolutions viewpoint and the classical Sigma-invariants that this paper extends to the locally compact setting.","marker":"[BR88]"},{"why":"Gives the homological finiteness criterion via essentially trivial reduced homology of Vietoris-Rips complexes that motivates Definition 5.2.","marker":"[Alo94]"},{"why":"Source of the classical homological and homotopical Sigma-invariants that Theorem C generalizes.","marker":"[Ren88]"},{"why":"Structure theorem for locally compact abelian groups used in the proof of Theorem K for abelian quotients.","marker":"[Mor77]"}],"fun_headline_variants":["Locally compact groups gain homological Σ-theory","Chain endomorphisms yield homological Σ for locally compact groups","Σ-invariant goes homological for locally compact groups","Homological Σ: the algebraic route for locally compact groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the companion paper's homotopical machinery is correct and that the newly introduced finitely modeled condition is the right finiteness notion for locally compact groups.","fun_headline_variants_meta":{"raw":{"variants":["Locally compact groups gain homological Σ-theory","Chain endomorphisms yield homological Σ for locally compact groups","Σ-invariant goes homological for locally compact groups","Homological Σ: the algebraic route for locally compact groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000904,"raw_usage":{"total_tokens":3909,"prompt_tokens":982,"completion_tokens":2927,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":2861}},"tokens_in":598,"tokens_out":2927,"duration_ms":25013,"temperature":1.0,"reasoning_tokens":2861,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:37:31.598440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\Sigma^2_{\\mathrm{top}}(G;\\mathbb Z)$ for the discrete group $G=\\mathrm{BS}(2,3)=\\langle a,b\\mid b^{-1}a^2b=a^3\\rangle$ directly from Definition 5.1 (where compact subsets are finite) and compare with the known classical $\\Sigma^2(G;\\mathbb Z)$; any disagreement would refute Theorem D, and the same computation tests whether the finitely modeled chain endomorphism criterion of Theorem F reproduces the same invariant.","supporting_citations":[],"review_version":1}