Cup and cap products for cohomology and homology groups of ample groupoids

Hiroki Matui; Takehiko Mori

REVIEW 3 major objections 4 minor 1 cited by

Cup and cap products for cohomology and homology groups of ample groupoids

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.14906 v3 pith:6P2NE6XH submitted 2024-11-22 math.OA math.DS

Hiroki Matui , Takehiko Mori This is my paper · ORCID

classification math.OAmath.DS MSC 22A2255N4546L4037B50

keywords cupproductcapamplegroupoidcohomologyhomologyPenrosetilingAmmannasymptoticallyinnerautomorphism

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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]).

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

A 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.

Who 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sections 5.3–5.4] 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.
[Sections 5.3–5.4, Theorems 5.4–5.5] 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.
[Proposition 4.2 and Theorem 4.5, proofs] 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.

minor comments (4)

[Section 3, Proposition 3.4] 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 4, Theorem 4.5 proof] 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 5.2] 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).
[Throughout, especially Section 5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cup and cap products are defined directly from the groupoid complexes, and the tiling computations are external simplicial calculations; the main caveat is an unproved ring-isomorphism identification, which is a correctness gap, not a circular reduction.

full rationale

The derivation chain in this paper is not circular. Section 3 defines the cup product ξ⌣η and cap product f⌢ξ directly on the groupoid chain and cochain complexes, and Theorems 3.2 and 3.5 are proved by elementary coboundary computations (Propositions 3.1 and 3.4) that do not assume the target statements. Remarks 3.3 and 3.6 only place this concrete model inside existing sheaf-theoretic frameworks of Moerdijk and Crainic-Moerdijk; they are not the basis of the well-definedness proofs. Section 4 imports Lin's theorem and the first author's earlier homology results [25] as external theorems; these citations carry independent content and are not used to define the products. The Penrose and Ammann computations (Sections 5.3 and 5.4) are carried out on the Anderson-Putnam complex Γ using data from [2], and Theorems 5.4 and 5.5 literally state products on H*(Γ). The advertised extension to H*(G) or the tiling space Ω relies on the assertion that H^i(Ω) ≅ H^i(G) ≅ H^i(Γ) is a ring isomorphism; the paper does not prove that the cup product is preserved by these identifications, and Remark 3.3 only addresses the isomorphism to H*(BG). That is an omitted proof or correctness gap, not a circular reduction: the groupoid cup product is defined independently, and the Γ cup product is computed from external cell-complex data, so neither reduces by construction to the paper's own inputs. No fitted parameter is renamed a prediction, and no uniqueness claim from the authors' prior work is used to force a choice.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper contains no fitted parameters or invented entities. All numerical data (cup product tables, basis elements, matrices) are outputs of computations, not inputs. The central claims rest on standard external theorems (Moerdijk's identification with sheaf cohomology, Lin's classification of asymptotically inner automorphisms, the gap labeling theorem for N ≤ 3, Matui's description of H1 via bisections, and Anderson-Putnam's inverse limit model for tiling spaces), which are cited but not proved in the paper.

assumptions (6)

domain assumption The cohomology of an etale groupoid G with constant sheaf Z is isomorphic to the cohomology of its classifying space BG, and the isomorphism intertwines cup products.
Invoked in Remark 3.3 and Sections 5.3-5.4; the paper cites Moerdijk [28] for the cohomology isomorphism and then identifies the cup products without proving ring compatibility.
domain assumption For the Penrose and Ammann substitution tilings, the tiling space Ω is homeomorphic to the inverse limit of the Anderson-Putnam complex (Γ,γ), and the projection induces isomorphisms on cohomology that preserve cup products.
Used in Sections 5.3 and 5.4 to replace H^*(G) with H^*(Γ); the cup product ring structure of the tiling space is inherited from Γ through the inverse limit projection.
domain assumption The gap labeling theorem holds for free minimal actions of Z^N on the Cantor set for N ≤ 3, giving Im D_A = Im D_G.
The proof of Proposition 4.2 needs Im D_A = Im D_G; Remark 4.3 explains that general proofs have issues, hence the restriction to N ≤ 3.
standard math Lin's theorem ([22, Corollary 11.2]): a unital separable simple C*-algebra with tracial rank zero and UCT has an automorphism asymptotically inner iff KK(α)=KK(id) and the rotation map vanishes.
Applied in Theorem 4.5 to reduce asymptotic innerness to vanishing of KK-difference and rotation map.
domain assumption For free minimal Z^N-actions, the transformation groupoid is almost finite, and every element of H1(G) is represented by a bisection (Matui [25, Lemma 6.3, Theorem 7.5]).
Used in the proofs of Proposition 4.2 and Theorem 4.5 to represent homology classes as unitaries 1_U obtained from bisections.
domain assumption The cycle f in Section 5.2 induces Poincare duality isomorphisms [f]⌢·: H^i(G) → H^{N-i}(G) for Z^N actions.
Used in Section 5.2 as a 'well-known' fact to express products in terms of the cocycle components.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Cup and cap products for cohomology and homology groups of ample groupoids." pith.science (2026). https://pith.science/paper/6P2NE6XH

@misc{pith2026241114906,
  author       = {Pith},
  title        = {Pith review of: Cup and cap products for cohomology and homology groups of ample groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6P2NE6XH}},
  note         = {Machine review of arXiv:2411.14906}
}

abstract

This paper explores the cup and cap products within the cohomology and homology groups of ample groupoids, focusing on their applications and fundamental properties. Ample groupoids, which are \'etale groupoids with a totally disconnected unit space, play a crucial role in the study of topological dynamical systems and operator algebras. We introduce the cup product, which defines a bilinear map on cohomology classes, providing a graded ring structure, and the cap product, which defines a bilinear map relating homology and cohomology. The paper aims to make these concepts accessible to a broader mathematical audience, offering clear definitions and detailed explanations. We also demonstrate an application of the cap product in the analysis of automorphisms of groupoid $C^*$-algebras. Specifically, we show how it helps determine the asymptotic innerness of automorphisms. Our results include the first explicit computations of cup products in the cohomology of tiling spaces, which may pave the way for new research in this area.

Figures

Figures reproduced from arXiv: 2411.14906 by the authors.

**Figure 1.** the cell complex Γ be the coboundary operators. Then, as computed in [2, Section 10.4], we get H1 (Γ) = Ker δ ′ 2/ Im δ ′ 1 ∼= Z 5 and H2 (Γ) = Z F / Im δ ′ 2 ∼= Z 8 . We regard Z E and Z F as free abelian groups generated by E and F, respectively. For n = 0, 1, 2, 3, 4, we define ξn ∈ Z E by ξn := an − bn+2 + cn+2 − dn+4 + en + en+2 − fn+2 − fn+4 − gn+3 − gn+4 + hn + hn+1. Define η ∈ Z E by η := X 4 n=0 (an + en − … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Cohomology of ample groupoids
math.OA 2024-12 accept novelty 4.0 of 10

The module-based cochain complex for ample groupoids computes the same cohomology groups as the standard continuous cocycle cohomology.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages · cited by 1 Pith paper

[2]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated C∗-algebras, Ergodic Theory Dynam. Systems, 18 (1998), pp. 509– 537

work page 1998
[1]

Akkermans, Y

E. Akkermans, Y. Don, J. Rosenberg, and C. L. Schochet, Relating diffraction and spectral data of aperiodic tilings: towards a Bloch theorem, J. Geom. Phys., 165 (2021), pp. Paper No. 104217, 23

work page 2021
[3]

Bellissard, R

J. Bellissard, R. Benedetti, and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling , Comm. Math. Phys., 261 (2006), pp. 1–41

work page 2006
[4]

Bellissard, J

J. Bellissard, J. Kellendonk, and A. Legrand, Gap-labelling for three-dimensional aperiodic solids, C. R. Acad. Sci. Paris S´ er. I Math., 332 (2001), pp. 521–525

work page 2001
[5]

Benameur and H

M.-T. Benameur and H. Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard) , in Operator algebras and mathematical physics (Constant ¸a, 2001), Theta, Bucharest, 2003, pp. 11–22. 21

work page 2001
[6]

B¨ onicke, C

C. B¨ onicke, C. Dell’Aiera, J. Gabe, and R. Willett, Dynamic asymptotic dimension and Matui’s HK conjecture , Proc. Lond. Math. Soc. (3), 126 (2023), pp. 1182–1253

work page 2023
[7]

Crainic, Cyclic cohomology and characteristic classes for foliations , Ph.D

M. Crainic, Cyclic cohomology and characteristic classes for foliations , Ph.D. thesis, Universiteit Utrecht, (2000)

work page 2000
[8]

A Homology Theory for Etale Groupoids

M. Crainic and I. Moerdijk, A homology theory for ´ etale groupoids , preprint. arXiv:math/9905011

work page Pith review arXiv

Show all 38 references

[9]

Crainic and I

M. Crainic and I. Moerdijk, A homology theory for ´ etale groupoids, J. Reine Angew. Math., 521 (2000), pp. 25–46

2000
[10]

de la Harpe and G

P. de la Harpe and G. Skandalis, D´ eterminant associ´ e ` a une trace sur une alg´ ebre de Banach, Ann. Inst. Fourier (Grenoble), 34 (1984), pp. 241–260

1984
[11]

Farsi, A

C. Farsi, A. Kumjian, D. Pask, and A. Sims, Ample groupoids: equivalence, homology, and Matui’s HK conjecture , M¨ unster J. Math., 12 (2019), pp. 411–451

2019
[12]

G¨ ahler, J

F. G¨ ahler, J. Hunton, and J. Kellendonk, Integral cohomology of rational projection method patterns, Algebr. Geom. Topol., 13 (2013), pp. 1661–1708

2013
[13]

Godement, Topologie alg´ ebrique et th´ eorie des faisceaux, vol

R. Godement, Topologie alg´ ebrique et th´ eorie des faisceaux, vol. XIII of Publications de l’Institut de Math´ ematique de l’Universit´ e de Strasbourg, Hermann, Paris, 1973. Troisi` eme ´ edition revue et corrig´ ee, Actualit´ es Scientifiques et Industrielles, No. 1252. [C...

1973
[14]

Haefliger, Differential cohomology , in Differential topology (Varenna, 1976), Liguori, Naples, 1979, pp

A. Haefliger, Differential cohomology , in Differential topology (Varenna, 1976), Liguori, Naples, 1979, pp. 19–70

1976
[15]

Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002

A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002

2002
[16]

Kaminker and I

J. Kaminker and I. Putnam, A proof of the gap labeling conjecture , Michigan Math. J., 51 (2003), pp. 537–546

2003
[17]

Kellendonk, Noncommutative geometry of tilings and gap labelling , Rev

J. Kellendonk, Noncommutative geometry of tilings and gap labelling , Rev. Math. Phys., 7 (1995), pp. 1133–1180

1995
[18]

Kellendonk and I

J. Kellendonk and I. F. Putnam, Tilings, C∗-algebras, and K-theory, in Directions in mathematical quasicrystals, vol. 13 of CRM Monogr. Ser., Amer. Math. Soc., Providence, RI, 2000, pp. 177–206

2000
[19]

Kerr and P

D. Kerr and P. Naryshkin, Elementary amenability and almost finiteness , preprint. arXiv:2107.05273

arXiv
[20]

Kishimoto and A

A. Kishimoto and A. Kumjian, The Ext class of an approximately inner automor- phism. II , J. Operator Theory, 46 (2001), pp. 99–122

2001
[21]

Li, A new approach to recent constructions of C∗-algebras from modular index theory, J

X. Li, A new approach to recent constructions of C∗-algebras from modular index theory, J. Funct. Anal., 269 (2015), pp. 841–864

2015
[22]

Lin, Asymptotically unitary equivalence and asymptotically inner automorphisms , Amer

H. Lin, Asymptotically unitary equivalence and asymptotically inner automorphisms , Amer. J. Math., 131 (2009), pp. 1589–1677. 22

2009
[23]

Matui, Ext and OrderExt classes of certain automorphisms of C∗-algebras arising from Cantor minimal systems , Canad

H. Matui, Ext and OrderExt classes of certain automorphisms of C∗-algebras arising from Cantor minimal systems , Canad. J. Math., 53 (2001), pp. 325–354

2001
[24]

Matui, Classification of homomorphisms into simple Z-stable C∗-algebras, J

H. Matui, Classification of homomorphisms into simple Z-stable C∗-algebras, J. Funct. Anal., 260 (2011), pp. 797–831

2011
[25]

Matui, Homology and topological full groups of ´ etale groupoids on totally discon- nected spaces, Proc

H. Matui, Homology and topological full groups of ´ etale groupoids on totally discon- nected spaces, Proc. Lond. Math. Soc. (3), 104 (2012), pp. 27–56

2012
[26]

Matui, Topological full groups of one-sided shifts of finite type , J

H. Matui, Topological full groups of one-sided shifts of finite type , J. Reine Angew. Math., 705 (2015), pp. 35–84

2015
[27]

Matui, ´Etale groupoids arising from products of shifts of finite type , Adv

H. Matui, ´Etale groupoids arising from products of shifts of finite type , Adv. Math., 303 (2016), pp. 502–548

2016
[28]

Moerdijk, Proof of a conjecture of A

I. Moerdijk, Proof of a conjecture of A. Haefliger , Topology, 37 (1998), pp. 735–741

1998
[29]

Niu, Radius of comparison and mean topological dimension: Zd-actions, Canad

Z. Niu, Radius of comparison and mean topological dimension: Zd-actions, Canad. J. Math., 76 (2024), pp. 1240–1266

2024
[30]

Ortega, The homology of the Katsura-Exel-Pardo groupoid, J

E. Ortega, The homology of the Katsura-Exel-Pardo groupoid, J. Noncommut. Geom., 14 (2020), pp. 913–935

2020
[31]

Ortega and A

E. Ortega and A. Sanchez, The homology of the groupoid of the self-similar infinite dihedral group, Math. Scand., 128 (2022), pp. 255–277

2022
[32]

Ortega and E

E. Ortega and E. Scarparo, Almost finiteness and homology of certain non-free ac- tions, Groups Geom. Dyn., 17 (2023), pp. 77–90

2023
[33]

N. C. Phillips, A classification theorem for nuclear purely infinite simple C∗-algebras, Doc. Math., 5 (2000), pp. 49–114

2000
[34]

Proietti and M

V. Proietti and M. Yamashita, Homology and K-theory of dynamical systems I. Torsion-free ample groupoids, Ergodic Theory Dynam. Systems, 42 (2022), pp. 2630– 2660

2022
[35]

Renault, A groupoid approach to C∗-algebras, vol

J. Renault, A groupoid approach to C∗-algebras, vol. 793 of Lecture Notes in Mathe- matics, Springer, Berlin, 1980

1980
[36]

Renault, Cartan subalgebras in C∗-algebras, Irish Math

J. Renault, Cartan subalgebras in C∗-algebras, Irish Math. Soc. Bull., (2008), pp. 29– 63

2008
[37]

Tanner, Rigidity, generators and homology of interval exchange groups , preprint

O. Tanner, Rigidity, generators and homology of interval exchange groups , preprint. arXiv:2304.13691

arXiv
[38]

Yi, Homology and Matui’s HK conjecture for groupoids on one-dimensional solenoids, Bull

I. Yi, Homology and Matui’s HK conjecture for groupoids on one-dimensional solenoids, Bull. Aust. Math. Soc., 101 (2020), pp. 105–117. 23

2020

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [2]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated C∗-algebras, Ergodic Theory Dynam. Systems, 18 (1998), pp. 509– 537

work page 1998

[2] [1]

Akkermans, Y

E. Akkermans, Y. Don, J. Rosenberg, and C. L. Schochet, Relating diffraction and spectral data of aperiodic tilings: towards a Bloch theorem, J. Geom. Phys., 165 (2021), pp. Paper No. 104217, 23

work page 2021

[3] [3]

Bellissard, R

J. Bellissard, R. Benedetti, and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling , Comm. Math. Phys., 261 (2006), pp. 1–41

work page 2006

[4] [4]

Bellissard, J

J. Bellissard, J. Kellendonk, and A. Legrand, Gap-labelling for three-dimensional aperiodic solids, C. R. Acad. Sci. Paris S´ er. I Math., 332 (2001), pp. 521–525

work page 2001

[5] [5]

Benameur and H

M.-T. Benameur and H. Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard) , in Operator algebras and mathematical physics (Constant ¸a, 2001), Theta, Bucharest, 2003, pp. 11–22. 21

work page 2001

[6] [6]

B¨ onicke, C

C. B¨ onicke, C. Dell’Aiera, J. Gabe, and R. Willett, Dynamic asymptotic dimension and Matui’s HK conjecture , Proc. Lond. Math. Soc. (3), 126 (2023), pp. 1182–1253

work page 2023

[7] [7]

Crainic, Cyclic cohomology and characteristic classes for foliations , Ph.D

M. Crainic, Cyclic cohomology and characteristic classes for foliations , Ph.D. thesis, Universiteit Utrecht, (2000)

work page 2000

[8] [8]

A Homology Theory for Etale Groupoids

M. Crainic and I. Moerdijk, A homology theory for ´ etale groupoids , preprint. arXiv:math/9905011

work page Pith review arXiv