{"id":"ba5a2679-4928-4daf-bb0e-98603028017d","arxiv_id":"2411.14906","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Concrete cup and cap product formulas for ample groupoids are developed and applied to tiling-space cohomology computations and to an asymptotic-innerness invariant for C*-algebra automorphisms.","lead":"This paper gives explicit formulas for cup and cap products in the cohomology and homology of ample groupoids, and uses them to compute new invariants and examples. It provides the first explicit cup product computations for Penrose and Ammann tiling cohomology, and a criterion for when automorphisms of groupoid C*-algebras are asymptotically inner.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Cup tables for Penrose/Ammann rely on an unproved ring-isomorphism from groupoid cohomology to the simplicial cohomology of the Anderson-Putnam complex; a sign or identification error would change the advertised results.","rationale":"The central definitions in Section 3 are internally sound; I checked the Leibniz rule in Proposition 3.1 and the sign in Proposition 3.4 against a simple group example, and the well-definedness arguments in Theorems 3.2 and 3.5 are correct. The SFT computation in Section 5.1 is consistent with the cap product formula. The remaining risk is the unproved identification of ring structures in the tiling applications; the paper itself flags related gaps only in Remark 4.3 (gap labelling), not here. The reader's weakest_assumption identifies exactly this ring-transfer step, and I agree. My own check of the Penrose cup table against the listed relations shows the Γ-level computation is internally consistent, so the issue is specifically the transfer to H^*(G), not arithmetic in Section 5.3. Since this affects the headline application but not the core product constructions, the appropriate verdict remains CONDITIONAL rather than REJECT or ACCEPT.","tokens_in":19261,"tokens_out":20935,"duration_ms":178045,"concrete_test":"Compute the Section 3 cup product directly on the tiling groupoid: express the five basis classes of H^1(G) from Theorem 5.4 (ξ_0,...,ξ_3,η) as explicit 1-cocycles on G^(1) (pattern-equivariant functions on the Cantor set), evaluate the wedge on lifts of the 2-cells to G^(2), and reduce the resulting 2-cocycles in H^2(G). Compare the resulting products to Theorem 5.4 modulo the chosen basis. Agreement confirms the ring transfer; any sign or basis mismatch would show the transfer fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.3 asserts that for the Penrose tiling groupoid G, H^i(G) is naturally isomorphic to H^i(Γ) and concludes \"Thus, it suffices to compute the cup product H^1(Γ) × H^1(Γ) → H^2(Γ).\" Section 5.4 repeats this for the Ammann tiling. The group-level isomorphisms are standard, but the paper gives no proof that this isomorphism (or the intermediate isomorphism H^*(G) ≅ H^*_Čech(Ω)) is a ring homomorphism for the Section 3 cup product. Remark 3.3 only says the groupoid cup product matches the sheaf cohomology cup product on the classifying space BG; it does not identify the tiling-space isomorphism used here with that sheaf-theoretic map, nor does it address the direct-limit structure. The advertised Theorems 5.4 and 5.5 therefore compute the ring of Γ, not provably the ring of G. If the transfer differs by a sign or a non-unital identification, the tables would not describe H^*(G).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces cup and cap products on the cohomology and homology of ample groupoids with constant coefficients. The cup product is defined on cochains and shown to be well-defined, giving a graded ring structure on H^*(G,Z); the cap product is defined on chains and cochains and shown to be well-defined. The paper also establishes basic associativity and functoriality properties, applies the cap product to the study of asymptotically inner automorphisms of groupoid C*-algebras arising from free minimal Z^N actions, and computes explicit examples: a cap product for SFT groupoids, a formula for Z^N actions, and cup product tables for the Penrose and Ammann tiling spaces using the Anderson–Putnam cell complexes.","tokens_in":19461,"tokens_out":15840,"duration_ms":145084,"significance":"If the ring-compatibility issues identified below are resolved, this paper provides a useful and accessible concrete model for cup and cap products in ample groupoid cohomology and homology, together with what would be the first explicit computations of cup products in tiling cohomology. The algebraic core, Theorems 3.2 and 3.5, is proved directly from the chain and cochain complexes without external machinery or fitted parameters, and the cap-product invariant for automorphisms is a clean application connecting groupoid homology to the de la Harpe–Skandalis determinant and Lin's asymptotic innerness criterion. The paper is careful to flag the current status of the gap labelling conjecture, which is an important sign of scholarly care.","major_comments":[{"comment":"The claim that 'it suffices to compute the cup product H^1(Γ) × H^1(Γ) → H^2(Γ)' presupposes that the isomorphisms H^*(G) ≅ H^*_Čech(Ω) ≅ H^*(Γ) are ring isomorphisms for the cup product defined in Section 3. The paper only cites group-level isomorphisms from [2] and does not prove or cite cup-product compatibility for the map from groupoid cohomology to the tiling-space cohomology; Remark 3.3 concerns H^*(BG), not H^*_Čech(Ω). Consequently, Theorems 5.4 and 5.5 as stated are theorems about the CW-complex Γ, and the advertised computation of the cup product on the cohomology of the tiling groupoid (or tiling space) is not established. Please add a proof or a precise reference for the ring compatibility, or revise the statements and surrounding text to make clear that the tables are for the Anderson–Putnam complex only.","section":"Sections 5.3–5.4"},{"comment":"The cup product tables contain only two worked products for the Penrose case and none for the Ammann case; all other entries are asserted via 'Other products can be computed in the same way.' Because the advertised novelty is the explicit computation, the omitted derivations make the tables unverifiable from the text. Please provide a complete set of intermediate products, including the face identifications used, or include a supplementary computational file.","section":"Sections 5.3–5.4, Theorems 5.4–5.5"},{"comment":"The proofs state that any element of H1(G) is represented by a cycle of the form 1_U for a single bisection U. Matui's theorem cited here ([25, Theorem 7.5]) gives generation by such classes, not individual representation; for a free Z^2 action, H1(G) ≅ Z^2 and a general class is a sum of multiples of bisection classes. The argument can be repaired by checking the invariant on the generating bisections and using linearity, and the wording 'every element' in the proof of Theorem 4.5 should be adjusted accordingly.","section":"Proposition 4.2 and Theorem 4.5, proofs"}],"minor_comments":[{"comment":"The displayed final equality has an unmatched parenthesis: it should read ∂_{n-m}(f ⌢ ξ) = (-1)^m((∂_n f ⌢ ξ) - (f ⌢ δ_m ξ)).","section":"Section 3, Proposition 3.4"},{"comment":"The phrase 'Since Z2 is free' should read 'Since Z^2 is free abelian' (or the action group is free abelian) to justify lifting the cocycle.","section":"Section 4, Theorem 4.5 proof"},{"comment":"The target of the Poincaré-duality map written as 'H^{N-i}(G)' should be the homology group H_{N-i}(G), since it is defined by capping with a class in H_N(G,Z).","section":"Section 5.2"},{"comment":"The paper uses the same symbol H with super/subscripts in OCR-unfriendly ways; in a final typeset version, ensure homology and cohomology groups are visually distinguished and that notation like H^1(Γ)×H^1(Γ)→H^2(Γ) is not confused with the groupoid groups in the preamble of Sections 5.3–5.4.","section":"Throughout, especially Section 5"}],"recommendation":"major_revision","confidential_remarks":"The algebraic core, Theorems 3.2 and 3.5, is sound and well presented, and the application to automorphisms is a genuine use of the cap product. The main issue is the missing ring-compatibility argument for the tiling-space identification; this is likely standard in the tiling cohomology literature, so the revision burden is moderate. The paper would also benefit from a more complete presentation of the tiling computations, since the 'Other products can be computed in the same way' sentences are not enough for the advertised first computations. I do not see grounds for rejection if the authors supply the missing justifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of the paper is genuine and in good shape. Theorems 3.2 and 3.5 give direct, checkable formulas for cup and cap products on constant-sheaf cohomology/homology of ample groupoids, and the proofs are short and standard. The paper is also honest about prior work: Remarks 3.3 and 3.6 explicitly credit Crainic and Moerdijk's sheaf-theoretic construction and position the contribution as a concrete, accessible model. That is the right framing. Proposition 3.7 (cap-cup associativity) and the functoriality results are useful and correctly proved.\n\nThe SFT computation in Theorem 5.2 is clean and fully derived, and the automorphism application in Section 4 is interesting. The use of the cap product to detect asymptotic innerness is a nice idea, and the gap-labelling caveat for N <= 3 is stated honestly (Remark 4.3), which I appreciate.\n\nThe soft spots are concentrated in the tiling sections. The stronger issue is the ring-transfer step in 5.3 and 5.4. The paper asserts that H*(G) is naturally isomorphic to H*(Gamma) and then says 'it suffices to compute the cup product' on Gamma. That suffices only if the isomorphism is a ring isomorphism, and no proof is given. Remark 3.3 compares the groupoid cup product to sheaf cohomology on the classifying space, but the identification used here goes through the tiling space and inverse limit, and that is not shown to preserve the product. The tables in Theorems 5.4 and 5.5 are therefore, strictly speaking, the cup product of the Anderson-Putnam complex, not provably of H*(G). I suspect the transfer is true, but the paper should either prove it or state it as an explicit assumption. As is, any reader who wants to use those tables has homework to do. This is a conditional-accept issue, not a fatal one.\n\nA smaller annoyance: several products are dismissed with 'Other products can be computed in the same way,' so the advertised tables are only partially derived. That is acceptable for a first pass but should be tightened in revision.\n\nWho is this for? People working in groupoid homology, C*-algebra dynamics, and tiling cohomology all get something from it. The main theorems are worth having in the literature in this explicit form. Send it to a serious referee. The tiling-transfer gap is the thing to ask the referee to pin down.","headline":"Solid, clearly-written constructions of cup/cap products for ample groupoids, with an honest but unproved ring-transfer step in the tiling computations; worth refereeing.","tokens_in":20016,"tokens_out":943,"would_cite":true,"duration_ms":11150,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A22","55N45","46L40","37B50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any ample groupoid, the cup product makes integer cohomology a graded ring, and the paper computes the resulting multiplication tables for the Penrose and Ammann tiling spaces for the first time.","keywords":["cup product","cap product","ample groupoid","groupoid cohomology","groupoid homology","Penrose tiling","Ammann tiling","asymptotically inner automorphism"],"falsifier":"For a single pair of 1-cocycles, say ξ0 and ξ1 of the Penrose construction, compute [ξ0]⌣[ξ1] ∈ $H^{2}$(G) directly from the Section 3 cochain formula using compact open bisections of the groupoid, and check whether it equals the class [A0]+[A1]−[A2]−[A3]+[E0]+[F0] predicted by Theorem 5.4 after translating through $H^{2}$(G)≅$H^{2}$(Γ); a mismatch in this one class would show the ring structures are not preserved by the identification.","tokens_in":19041,"feed_emoji":"🧩","tokens_out":8981,"duration_ms":83887,"temperature":0.7,"pith_summary":"Matui and Mori build well-defined cup and cap products for the homology and cohomology of ample groupoids, the étale groupoids with totally disconnected unit spaces that underlie Cantor-system and tiling C*-algebras. The cup product turns the integer cohomology H*(G,Z) into a graded ring, and the cap product pairs homology with cohomology. They show the cap product detects whether an automorphism of the associated groupoid C*-algebra is asymptotically inner, giving a criterion in terms of an invariant built from the product. As the headline application, they produce the first explicit cup-product tables for the cohomology of the Penrose and Ammann tiling spaces, computed on the Anderson–Putnam cell complex. If the standard isomorphisms between groupoid cohomology and the Čech cohomology of the hull preserve ring structure, these tables give the full cohomology rings of the two tiling spaces.","feed_headline":"First cohomology ring tables for Penrose and Ammann tilings","feed_subtitle":"Groupoid cohomology gains a ring structure, and the new tables give tiling invariants additive groups miss.","key_machinery":"The load-bearing object is the explicit chain-level formula for the products on the groupoid's spaces of composable strings. The cup product splits a string (g1,...,gn+m) after the nth entry; the cap product sums over the first m entries of an n-string, leaving an (n−m)-string. Correctness relies on the face maps $d_i^{{(n)}}$ and the Leibniz-type identities of Propositions 3.1 and 3.4, which show that cup products of cocycles are cocycles, and that cap products of cycles with cocycles are cycles. For the tiling computations, the machinery switches to the Anderson–Putnam complex Γ: because the natural isomorphism H^*(G) ≅ H^*_{Čech}(Ω) ≅ H^*(Γ) is used to import the simplicial cup product, the tables in Theorems 5.4 and 5.5 are obtained by reading off products of 1-cochains on the cell complex.","core_discovery":"The paper defines the cup product (ξ ⌣ η)(g1,...,gn+m) = ξ(g1,...,gn)·η(gn+1,...,gn+m) and the cap product (f ⌢ ξ)(h1,...,hn−m) = Σ f(g1,...,gm,h1,...,hn−m)·ξ(g1,...,gm), and proves they descend to well-defined bilinear maps H^n(G,Z)×H^m(G,A)→$H^{{n+m}}$(G,A) and H_n(G,Z)×H^m(G,A)→H_{n−m}(G,A). It verifies associativity of the cup product and the relation f⌢(ξ⌣η) = (f⌢ξ)⌢η. For the cohomology of tiling spaces, it identifies H^*(G) with H^*(Γ) for the Anderson–Putnam complex Γ and computes the cup product $H^{1}$(Γ)×$H^{1}$(Γ)→$H^{2}$(Γ) explicitly: for the Penrose tiling, with the stated bases, the products are given by four families of identities, and for the Ammann tiling, by six identities. The cap product is then applied to show that for cocycles ξ:G→T lifting to R, vanishing of ρ_G(c⌢[ξ]) for all c∈H_1(G) is necessary for the automorphism α_ξ to be asymptotically inner when N≤3, and sufficient for free minimal Z²-actions with $H^{1}$(G) free.","pith_inferences":["If the ring isomorphism between groupoid cohomology and the Čech cohomology of the hull is automatic (as the additive isomorphisms are), the tables in Theorems 5.4 and 5.5 give the full cohomology ring of the Penrose and Ammann continua; these rings are finer invariants than the additive groups and could distinguish aperiodic hulls that share Betti numbers.","The cap-product criterion for asymptotic innerness is phrased through the de la Harpe–Skandalis determinant; the same pairing may extend verbatim to actions of Z^N with N>3 once the gap labelling conjecture—which the authors note has published proofs with known issues (Remark 4.3)—is settled for those cases.","The Anderson–Putnam computation is mechanical enough that the same recipe could generate cup-product tables for other substitution tilings (octagonal, pinwheel, and similar), allowing ring-theoretic comparison among substitution tiling spaces."],"forward_implications":["For every ample groupoid G, H*(G,Z) is a graded ring, so ring-theoretic invariants such as nilpotence and the full multiplication table become available alongside the additive cohomology groups.","The cap product gives a concrete obstruction to asymptotic innerness: for N ≤ 3, any cocycle ξ with α_ξ asymptotically inner must satisfy ρ_G(c⌢[ξ])=0 for all c∈H_1(G), and for free minimal Z²-actions with H^1(G) free this condition is also sufficient.","For one-sided SFT groupoids, the cap product ·⌢[ξ] on H_1→H_0 is realized as the map a ↦ a + Im(I−A^t) on Ker(I−A^t), tying the product directly to the adjacency matrix.","The Penrose tiling cohomology ring is now explicit: with H^1≅Z^5 and H^2≅Z^8, the cup product is given by the identities in Theorem 5.4, including [ξ_n]⌣[ξ_{n+1}] = [A0]+[A1]−[A2]−[A3]+[E0]+[F0].","The Ammann tiling cohomology ring is also explicit: with H^1≅Z^4 and H^2≅Z^6, Theorem 5.5 lists the products, beginning with [ξ1]⌣[ξ2] = 2([A]+[D]+[E]+[H])."],"supporting_citations":[{"why":"Supplies the Anderson–Putnam cell complex Γ and the isomorphisms H^i(Ω)≅H^i(Γ) used to compute the cup product tables for the Penrose and Ammann tilings.","marker":"[2]"},{"why":"Provides the foundational homology of ample groupoids, the representation of H_1 elements by bisections, and the SFT homology computation used in Section 5.","marker":"[25]"},{"why":"Renault's groupoid approach to C*-algebras supplies the cocycle-to-automorphism construction and the cohomological framework used throughout.","marker":"[35]"},{"why":"Crainic and Moerdijk's sheaf-theoretic construction of cup and cap products for étale groupoids is the abstract model that the paper's concrete formulas instantiate.","marker":"[8, 7]"},{"why":"Moerdijk's isomorphism between H*(G;A) and sheaf cohomology of the classifying space is used in Remark 3.3 to identify the new cup product with the sheaf product.","marker":"[28]"},{"why":"Lin's characterization of asymptotically inner automorphisms is the external criterion invoked in Theorems 4.5 to turn the cap-product condition into asymptotic innerness.","marker":"[22]"},{"why":"Bellissard–Kellendonk–Legrand's gap labelling result for three-dimensional aperiodic solids is used to identify Im D_A with Im D_G when N≤3.","marker":"[4]"}],"fun_headline_variants":["Cohomology rings of Penrose and Ammann tilings computed","Penrose and Ammann tilings gain cohomology ring structure","First ring tables for tiling cohomology from ample groupoids","Penrose and Ammann cohomology rings explicitly described"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tiling computations stand or fall on the claim that the new groupoid cup product agrees, under the standard identifications of cohomology groups, with the ordinary cup product on the Anderson–Putnam cell complex—if those identifications only preserve additive structure, the tables describe a different ring.","fun_headline_variants_meta":{"raw":{"variants":["Cohomology rings of Penrose and Ammann tilings computed","Penrose and Ammann tilings gain cohomology ring structure","First ring tables for tiling cohomology from ample groupoids","Penrose and Ammann cohomology rings explicitly described"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001195,"raw_usage":{"total_tokens":4976,"prompt_tokens":1038,"completion_tokens":3938,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":3864}},"tokens_in":654,"tokens_out":3938,"duration_ms":29732,"temperature":1.0,"reasoning_tokens":3864,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:44:34.464315+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a single pair of 1-cocycles, say ξ0 and ξ1 of the Penrose construction, compute [ξ0]⌣[ξ1] ∈ $H^{2}$(G) directly from the Section 3 cochain formula using compact open bisections of the groupoid, and check whether it equals the class [A0]+[A1]−[A2]−[A3]+[E0]+[F0] predicted by Theorem 5.4 after translating through $H^{2}$(G)≅$H^{2}$(Γ); a mismatch in this one class would show the ring structures are not preserved by the identification.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Anderson–Putnam cell complex Γ and the isomorphisms H^i(Ω)≅H^i(Γ) used to compute the cup product tables for the Penrose and Ammann tilings."},{"cited_title":"Matui, Homology and topological full groups of ´ etale groupoids on totally discon- nected spaces, Proc","cited_arxiv_id":null,"evidence_quote":"Provides the foundational homology of ample groupoids, the representation of H_1 elements by bisections, and the SFT homology computation used in Section 5."},{"cited_title":"Renault, A groupoid approach to C∗-algebras, vol","cited_arxiv_id":null,"evidence_quote":"Renault's groupoid approach to C*-algebras supplies the cocycle-to-automorphism construction and the cohomological framework used throughout."},{"cited_title":"Moerdijk, Proof of a conjecture of A","cited_arxiv_id":null,"evidence_quote":"Moerdijk's isomorphism between H*(G;A) and sheaf cohomology of the classifying space is used in Remark 3.3 to identify the new cup product with the sheaf product."},{"cited_title":"Lin, Asymptotically unitary equivalence and asymptotically inner automorphisms , Amer","cited_arxiv_id":null,"evidence_quote":"Lin's characterization of asymptotically inner automorphisms is the external criterion invoked in Theorems 4.5 to turn the cap-product condition into asymptotic innerness."},{"cited_title":"Bellissard, J","cited_arxiv_id":null,"evidence_quote":"Bellissard–Kellendonk–Legrand's gap labelling result for three-dimensional aperiodic solids is used to identify Im D_A with Im D_G when N≤3."}],"review_version":1}