{"id":"3274a181-113f-4781-a8cf-f66a6be283b4","arxiv_id":"2506.06503","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines equivariant bivariant periodic cyclic homology for ample groupoids and proves homotopy invariance, stability, excision, and a Green-Julg isomorphism.","lead":"This paper creates a new mathematical tool for measuring the shape of spaces with layered symmetries, called groupoids, and shows the tool behaves like classical homology theories. The motivation is topological dynamics, where the tool could eventually help compute K-theory and test Matui's HK conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Excision rests on Theorem 5.11, stated without proof; since excision is a headline property of HP^G, the paper is conditional until that proof is supplied or independently verified.","rationale":"The paper's central claim is that HP^G satisfies the same formal properties as group-equivariant periodic cyclic homology. The construction of HP^G in Definition 4.13 is coherent, and Sections 2–4 contain substantial verified groundwork: the equivalence between G-modules and C∞_c(G)-comodules, the monoidal structure, and the explicit operators on Ω_G(A) with Lemma 4.4. These are real independent supports. However, the paper explicitly leaves the key step of excision unproved: Theorem 5.11 is stated after saying the authors 'will be rather brief and only sketch the main strategy,' and no proof follows. Since excision is one of the three headline properties listed in the abstract and is used in both variables, the conditional verdict is appropriate. I do not claim the theorem is false; the gap is a missing proof obligation. The most load-bearing aspect is that Theorem 5.11 controls the relative X-complex; without it, Theorem 5.12 and the claimed excision property are not established. The reader also flags Theorem 4.12, which is another transferred proof obligation, but Theorem 5.11 is the sharper locus because it is explicitly identified as the key step and receives no argument at all. I set verdict_should_be to UNCHANGED because the reader already marked the paper CONDITIONAL for exactly this kind of gap.","tokens_in":41565,"tokens_out":19814,"duration_ms":191604,"concrete_test":"Provide a complete proof of Theorem 5.11 by adapting the excision argument from [30, §9] to the groupoid setting, or run a minimal computational check: take G the transformation groupoid of a finite group acting on a Cantor set, choose an admissible extension of G-algebras with a C∞-linear but non-equivariant splitting, and verify the six-term exact sequence of Theorem 5.12 for HP^G. If the sequence is not exact for such an example, Theorem 5.11 fails as stated; if it is exact, the missing proof is a gap that can be closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.3 introduces Theorem 5.11 as 'the key step in the proof of the excision theorem' and asserts that ρ : X_G(TK) → X_G(TE:TQ) is a homotopy equivalence, but no proof or proof sketch is provided. Excision in both variables (Theorem 5.12) is then derived from this single unproved statement. The gap is not merely cosmetic: the paper only assumes the extension in Theorem 5.12 is admissible over C∞_c(G0), and the claimed reduction to the G-equivariant splitting hypothesis of Theorem 5.11 via tensoring with KG and 'the same argument as in Lemma 5.6' is itself not demonstrated. If the relative X-complex X_G(TE:TQ) is not appropriately controlled, the six-term exact sequences of Theorem 5.12 need not follow, so the advertised excision property of HP^G is unsupported. This is a proof obligation rather than a known counterexample; the surrounding formalism (Lemma 4.4, the monoidal structure, and the definition of HP^G) is consistent with the claimed result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper constructs a bivariant equivariant periodic cyclic homology theory HP^G_* for actions of ample Hausdorff groupoids on complex algebras, following the Cuntz–Quillen approach and Voigt's earlier group-equivariant theory. The authors introduce G-modules, G-algebras, G-anti-Yetter-Drinfeld modules, equivariant differential forms, the equivariant X-complex, and then state the main structural properties: homotopy invariance, stability, excision in both variables, and a Green–Julg type isomorphism for proper groupoids with paracompact quotient. The paper contains substantial new formalism for groupoids, especially the comodule description of G-modules and the monoidal tensor product. However, two results that carry much of the theory are not proved: Theorem 4.12, the homotopy equivalence between the X-complex of the periodic tensor algebra and the Hodge tower, is dismissed as a direct translation of [30, Theorem 8.6], and Theorem 5.11, the key step in excision, is stated without proof or proof sketch. The paper is therefore conditional in its present form.","tokens_in":41786,"tokens_out":5411,"duration_ms":58820,"significance":"If the main results hold, this is a valuable contribution: it provides a bivariant periodic cyclic homology for groupoid actions with the same formal properties as the group-equivariant theory, and a Green–Julg theorem that could be relevant to the Baum–Connes approach to Matui's HK conjecture. The paper gives detailed, self-contained arguments for several foundational pieces, including the equivalence between G-modules and C_c^∞(G)-comodules (Proposition 3.6), the monoidal structure (Proposition 3.8), the paracomplex identities (Lemma 4.4), and the stability theorem (Theorem 5.5). These are genuine assets. The main weakness is not the overall strategy but the number of load-bearing assertions that are currently left unproved; the advertised theory and its headline properties are not yet fully supported as written.","major_comments":[{"comment":"Theorem 4.12 states that X_G(TA) and the Hodge tower θΩ_G(A) are homotopy equivalent as pro-paracomplexes of G-anti-Yetter-Drinfeld modules, but the proof is omitted with the comment that it is a direct translation of [30, Theorem 8.6]. This is not satisfactory in the present setting: the groupoid case changes the category of coefficient modules, the operator T, and the definitions of b_G and B_G, so it is not formally the same statement as in the group case. The authors should either give the full proof or, at minimum, indicate precisely which arguments of [30, §8] survive without change and which require groupoid-specific verification. As written, this central structural result is unsupported.","section":"§4.5, Theorem 4.12"},{"comment":"Theorem 5.11 is introduced as 'the key step in the proof of the excision theorem' and asserts that ρ : X_G(TK) → X_G(TE:TQ) is a homotopy equivalence, but no proof or proof sketch is provided. Since Theorems 5.12 and the advertised excision property in both variables are direct consequences of this single assertion, the paper's central claim about excision is currently unproved. The authors need to supply the full argument or a detailed reduction to the group-equivariant case that accounts for the groupoid-specific X-complex and AYD module structure.","section":"§5.3, Theorem 5.11"},{"comment":"Even assuming Theorem 5.11, the reduction from an extension that is admissible only over C_c^∞(G^(0)) to one satisfying the G-equivariant splitting hypothesis of Theorem 5.11 is not demonstrated. The text says that tensoring with K_G and 'the same argument as in the proof of Lemma 5.6' gives the required G-equivariant pro-linear splitting, but this is not explained. Lemma 5.6 concerns the construction of a G-linear isomorphism from a C_c^∞(G^(0))-linear isomorphism; it does not by itself produce a G-equivariant splitting of an algebra extension with the required properties. This gap is load-bearing for the excision theorem.","section":"§5.3, proof of Theorem 5.12"},{"comment":"The proof of the Green–Julg theorem relies on several unverified localisation identifications, and in particular on the assertion that the map HH_*(γ_G) 'agrees up to a nonzero scalar' with the known isomorphism from the finite-group Green–Julg theorem. Since Green–Julg is one of the headline results, these identifications should be spelled out in enough detail to be checked, or the relevant statements should be isolated as explicit lemmas with proofs. As written, the argument is too schematic at exactly the point where the groupoid case is reduced to the group case.","section":"§6, proof of Theorem 6.6"}],"minor_comments":[{"comment":"There is a typo in 'This s convenient when it comes to discussing quasifreeness'; it should read 'This is convenient'.","section":"§1, page 3"},{"comment":"The text refers to 'Theorem 3.6' in 'Theorem 3.6 can be phrased as saying...', but the statement is Proposition 3.6; the cross-reference should be corrected.","section":"§3.3, after Proposition 3.6"},{"comment":"The term 'G-lonilcur' is used without definition. If it is an abbreviation for 'locally nilpotent curvature', it should be introduced explicitly before first use.","section":"§4.1, page 24"},{"comment":"The notation E^∞ and C_c^∞(G^(0))^∞ is introduced informally; the direct sum pairing on these modules and the sense in which the isometric isomorphism is induced by Lemma 5.7 should be stated more explicitly.","section":"§5.2, proof of Theorem 5.8"},{"comment":"The Hom-complex in the definition is written as Hom_{A(G)}, but the crossed product A(G)=O_G ⋊ G is only described verbally in §3.4; it would improve readability to restate this identification at the point of definition.","section":"§4.6, Definition 4.13"}],"recommendation":"major_revision","confidential_remarks":"This is a serious manuscript with substantial new formalism, and the missing proofs are likely fillable rather than signs of a false result. However, the paper currently leaves two load-bearing assertions - Theorem 4.12 and especially Theorem 5.11 - without proof, and the reduction in the proof of Theorem 5.12 is also under-explained. I would not reject the paper, but I would not recommend acceptance until these proof obligations are met. The heavy reliance on [30] is legitimate, but 'direct translation' is too strong a claim for statements in a new algebraic and groupoid setting; the authors should be asked to provide the relevant details or precise references to where each adaptation is proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere's the paper. It defines bivariant equivariant periodic cyclic homology for actions of ample Hausdorff groupoids, with a construction that genuinely generalizes Voigt's group-equivariant theory. That's new: previous equivariant theories stopped at discrete groups. The comodule presentation of G-modules and the Green-Julg analogue for proper groupoids are also new. The paper is honest about its debts, and the sections with full proofs—monoidal structure, homotopy invariance, stability—look coherent. I checked the formulas in Lemma 4.4 and the stability proof closely enough to believe they work.\n\nBut the paper as written is conditional. Two load-bearing results are not proved. Theorem 4.12 (X_G(TA) homotopy equivalent to the Hodge tower) is dismissed as a direct translation of [30, Theorem 8.6]. That's plausible, since the groupoid formalism was set up to make the proof translate, but it is still an unproved assertion at the base of the definition. More serious is Theorem 5.11, the key step in excision: rho : X_G(TK) -> X_G(TE:TQ) is stated as a homotopy equivalence with no proof, and excision in both variables is derived from it. The stress-test note is right that the extension in Theorem 5.12 is only admissible over C_c^infty(G0), and the reduction to the G-equivariant splitting hypothesis via tensoring with KG is not demonstrated. This is a proof obligation, not a known counterexample—I don't see an error in the surrounding formalism—but excision is one of the three headline properties in the abstract, so the advertised theorem currently rests on an assertion.\n\nThe Green-Julg proof has a similar flavor: it localizes to finite groups and then imports the finite-group result, with a \"one checks\" up to scalar at the critical comparison. I believe it's probably right, but again the detail level is not enough for independent verification.\n\nMy overall read: the construction is significant, the parts that are proved seem correct, and the missing proofs are likely fillable. I'd send this to a serious referee, but the referee should be told that Theorems 4.12 and 5.11 need complete proofs or explicit statements of what is being taken from the group case. It's not desk-reject material; it's revise-with-proofs material.","headline":"This is the first equivariant periodic cyclic homology for ample groupoids; the construction is new and largely credible, but excision and the periodic tensor algebra step rest on unproved assertions that should be supplied before the headline claims are accepted.","tokens_in":42290,"tokens_out":2496,"would_cite":true,"duration_ms":24398,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D55","16E40","22A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines bivariant equivariant periodic cyclic homology for actions of ample groupoids and proves homotopy invariance, stability, and excision in both variables, together with a Green-Julg isomorphism for proper groupoids.","keywords":["equivariant periodic cyclic homology","ample groupoids","bivariant homology theory","Green-Julg theorem","excision","homotopy invariance","stability","anti-Yetter-Drinfeld modules"],"falsifier":"Take a proper ample groupoid with a nontrivial finite stabiliser, for instance a finite group acting on a Cantor set, and compute both sides of the Green-Julg isomorphism of Theorem 6.1 for $A=C^\\infty_c(G^{(0)})$: if the equivariant periodic cyclic homology of $A$ and the periodic cyclic homology of $A\\rtimes G$ differ in degree 0 or 1, the theorem is false.","tokens_in":41307,"feed_emoji":"🔄","tokens_out":15192,"duration_ms":141464,"temperature":0.7,"pith_summary":"Bivariant equivariant periodic cyclic homology is constructed for actions of an ample Hausdorff groupoid $G$ on complex algebras: for pro-$G$-algebras $A$ and $B$ the paper defines groups $HP^G_*(A,B)$ from the equivariant $X$-complex of the stabilised periodic tensor algebra. It establishes that $HP^G_*$ is homotopy invariant under smooth $G$-equivariant homotopies, stable under tensoring with algebras of finite-rank operators on $G$-modules with invariant pairings, and excisive in both variables, matching the formal behaviour of the discrete-group theory. In the proper case, with $G/G^{(0)}$ paracompact, it proves a Green-Julg isomorphism comparing the theory with the periodic cyclic homology of the crossed product. The motivation is to produce a computable invariant for the $K$-theory of ample groupoids in the direction of the HK-conjecture.","feed_headline":"New equivariant cyclic homology is homotopy invariant and excisive","feed_subtitle":"Built for ample groupoids, with stability and a Green-Julg isomorphism for proper groupoids.","key_machinery":"The central object is the equivariant $X$-complex $X_G(A)$, a two-term pro-paracomplex of $G$-anti-Yetter-Drinfeld modules — $G$-modules with a compatible action of the algebra of functions on the loop groupoid — whose differential squares to $\\mathrm{id}-T$, where $T$ is the canonical automorphism coming from the adjoint action. Because the square is not zero, one must pass to Hom-complexes or mapping complexes to get honest chain complexes; this is why the paper defines bivariant periodic cyclic homology rather than ordinary cyclic homology. The argument is carried by the periodic tensor algebra $TA$ and the Hodge tower $\\theta\\Omega_G(A)$, together with the assertion that $X_G(TA)$ is homotopy equivalent to $\\theta\\Omega_G(A)$, which lets paracomplex homotopy equivalences become isomorphisms in $HP^G_*$. Stability is proved through twisted traces from $G$-equivariant pairings, excision through a key homotopy equivalence between $X_G(TK)$ and the kernel of $X_G(TE)\\to X_G(TQ)$, and the Green-Julg theorem through localisation at orbits and an averaging map that reduces to finite stabilisers.","core_discovery":"On the paper's own terms, for every ample Hausdorff groupoid $G$ and every pair of pro-$G$-algebras $(A,B)$, the groups $HP^G_*(A,B)$ are defined by taking the Hom-complex of the equivariant $X$-complexes of the stabilised periodic tensor algebras, and the main claim is that this is a bivariant homology theory. Smooth $G$-equivariant homotopies induce the same map in degree zero; tensoring with a $G$-module $E$ with a $G$-equivariant pairing gives an invertible class; and every admissible extension of pro-$G$-algebras yields six-term exact sequences in both variables. For proper $G$ with $G/G^{(0)}$ paracompact, the Green-Julg isomorphism $HP^G_*(C^\\infty_c(G^{(0)}),A) \\cong HP^{C^\\infty_c(G/G^{(0)})}_*(C^\\infty_c(G/G^{(0)}), A\\rtimes G)$ holds. When the unit space is a point, the construction reduces to the earlier group-equivariant periodic cyclic homology.","pith_inferences":["Inference: if the two omitted groupoid versions of the group-case equivalences (Theorem 4.12 and Theorem 5.11) hold as stated, the same construction should extend to bornological vector spaces almost unchanged, giving analytic versions for etale groupoids; the paper notes this but does not develop it.","Inference: a bivariant Chern character combined with the Green-Julg isomorphism could yield a new comparison between $HP^G_*$ and groupoid homology in the direction of the HK-conjecture; this connection is not drawn in the paper.","Inference: a concrete next test is to compute $HP^G_*$ for a minimal non-discrete ample groupoid such as the transformation groupoid of a full shift and compare the result with groupoid homology; agreement would support, and disagreement constrain, the conjectural link.","Inference: the authors' suggestion that Hausdorff covers of non-Hausdorff groupoids might repair the failure of locally constant functions to multiply indicates a plausible route to extending $HP^G_*$ beyond Hausdorff groupoids, but no construction is given."],"forward_implications":["Admissible extensions of pro-$G$-algebras produce six-term exact sequences in both variables, so $HP^G_*$ can be used as a two-variable homological invariant in the same way as bivariant $K$-theory.","For proper ample groupoids with paracompact quotient, $HP^G_*(C^\\infty_c(G^{(0)}),A)$ is computable from the crossed product $A\\rtimes G$, a substantial simplification for concrete examples.","Stability means finite-rank perturbations and smoothing operators can be introduced without changing the theory, provided the $G$-module carries an admissible invariant pairing.","For discrete groupoids the theory decomposes as a direct sum over orbits of stabiliser-equivariant periodic cyclic homology, reducing computations to smaller groups.","The theory carries a module structure over the conjugation-invariant functions on the loop groupoid, and localising at an ideal isolates contributions from individual conjugacy classes, which is useful for Chern character calculations."],"supporting_citations":[{"why":"Supplies the equivariant $X$-complex, the Hodge tower, and the group-equivariant theorem that Theorem 4.12 translates to groupoids.","marker":"[30]"},{"why":"Provides the group-equivariant cyclic homology theory and the finite-group Green-Julg comparison used in the localisation argument.","marker":"[29]"},{"why":"Establishes excision in bivariant periodic cyclic cohomology, the template for the excision argument in Section 5.3.","marker":"[15]"},{"why":"Introduces the periodic tensor algebra and quasifreeness machinery on which the definition of $HP^G_*$ is built.","marker":"[13]"},{"why":"Provides the Hodge tower and mixed-complex technology that turns paracomplex homotopy equivalences into homology isomorphisms.","marker":"[14]"},{"why":"Supplies the lonilcur-map and excision ideas used in the proof strategy for Section 5.","marker":"[20]"},{"why":"Gives the proper-groupoid setup, cut-off functions, and the Green-Julg theorem that the paper adapts to periodic cyclic homology.","marker":"[26]"},{"why":"Provides the finite-group Green-Julg isomorphism in Hochschild homology used at the end of Theorem 6.6.","marker":"[6]"},{"why":"Supplies the comparison between equivariant differential forms and crossed products for finite groups used in the same reduction.","marker":"[7]"}],"fun_headline_variants":["New cyclic homology for ample groupoids is excisive","Green-Julg theorem proven for proper groupoid actions","Equivariant cyclic homology: homotopy, stability, excision","Bivariant cyclic homology theory for ample groupoids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the two technical equivalences that make the definition and excision work in the discrete-group case still work for every ample groupoid; the paper states them without giving the groupoid proofs.","fun_headline_variants_meta":{"raw":{"variants":["New cyclic homology for ample groupoids is excisive","Green-Julg theorem proven for proper groupoid actions","Equivariant cyclic homology: homotopy, stability, excision","Bivariant cyclic homology theory for ample groupoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000128,"raw_usage":{"total_tokens":1049,"prompt_tokens":810,"completion_tokens":239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":426,"completion_tokens_details":{"reasoning_tokens":172}},"tokens_in":426,"tokens_out":239,"duration_ms":2641,"temperature":1.0,"reasoning_tokens":172,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:55:46.016359+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a proper ample groupoid with a nontrivial finite stabiliser, for instance a finite group acting on a Cantor set, and compute both sides of the Green-Julg isomorphism of Theorem 6.1 for $A=C^\\infty_c(G^{(0)})$: if the equivariant periodic cyclic homology of $A$ and the periodic cyclic homology of $A\\rtimes G$ differ in degree 0 or 1, the theorem is false.","supporting_citations":[{"cited_title":"Equivariant periodic cyclic homology.J","cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant $X$-complex, the Hodge tower, and the group-equivariant theorem that Theorem 4.12 translates to groupoids."},{"cited_title":"Equivariant cyclic homology.PhD thesis, University of M¨ unster, 2003","cited_arxiv_id":null,"evidence_quote":"Provides the group-equivariant cyclic homology theory and the finite-group Green-Julg comparison used in the localisation argument."},{"cited_title":"Excision in bivariant periodic cyclic cohomology.Invent","cited_arxiv_id":null,"evidence_quote":"Establishes excision in bivariant periodic cyclic cohomology, the template for the excision argument in Section 5.3."},{"cited_title":"Algebra extensions and nonsingularity.J","cited_arxiv_id":null,"evidence_quote":"Introduces the periodic tensor algebra and quasifreeness machinery on which the definition of $HP^G_*$ is built."},{"cited_title":"Cyclic homology and nonsingularity.J","cited_arxiv_id":null,"evidence_quote":"Provides the Hodge tower and mixed-complex technology that turns paracomplex homotopy equivalences into homology isomorphisms."},{"cited_title":"Analytic cyclic homology.PhD Thesis, University of M¨ unster, 1999","cited_arxiv_id":null,"evidence_quote":"Supplies the lonilcur-map and excision ideas used in the proof strategy for Section 5."},{"cited_title":"La conjecture de Novikov pour les feuilletages hyperboliques.K-Theory, 16(2):129–184, 1999","cited_arxiv_id":null,"evidence_quote":"Gives the proper-groupoid setup, cut-off functions, and the Green-Julg theorem that the paper adapts to periodic cyclic homology."},{"cited_title":"Algebras associated with group actions and their homology.Brown Uni- versity preprint, 1986","cited_arxiv_id":null,"evidence_quote":"Provides the finite-group Green-Julg isomorphism in Hochschild homology used at the end of Theorem 6.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the comparison between equivariant differential forms and crossed products for finite groups used in the same reduction."}],"review_version":1}