Eduard Ortega, Alvaro Sanchez
We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.
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We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu{Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain co…
Ample groupoid homology and étale correspondences
本文证明充裕群胚间的平展对应诱导同调群同态，并通过子群胚的归纳—限制伴随构造任意群胚模的消解，以 Tor 图景显式刻画相应同调映射。
Isomorphisms in K-theory from isomorphisms in groupoid homology theories
该研究证明无挠顺从充裕群胚中由平展对应诱导的群胚同调同构可导出相关$\text{C}^\text{*}$-代数K-理论同构，借此推广了X. Li的左正则逆半群$\text{C}^\text{*}$-代数K-理论公式，核心方法是发展ABC谱序列的函子性。
Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
本文研究étale群胚同调理论，证明了群胚乘积的Künneth公式和主群胚的Poincaré对偶型结果，并推广了幂零群系统及类螺线管系统的同调计算，证明了HK猜想。
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete isotropy groups, - G is Morita equivalent to an etale groupoid. Moreover, we show that among the Lie groupoids integrating a given foliation, the holonomy and the monodromy groupoids are extreme examples. The second theorem shows that the cyclic homology of convolution algebras …
We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.
We introduce a graded homology theory for graded étale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients $\mathbb Z$ is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading…
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.
Discretisation and independent resolutions of ample groupoids
提出基于分立化和独立分解的框架，通过将充分广群等价于离散广群，并构建独立分解，系统研究其同调与K理论，应用于Garside范畴等例子。
We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the $K$-theory of its groupoid $C^*$-algebra.
For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H_1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H_1 to K_1 of the groupoi…
Groupoid homology and K-theory for algebraic actions from number theory
计算代数整数环和整数动力学的ample群胚同调，推导拓扑全群的同调结果，并利用群胚同调计算环C*-代数的K理论，解决了Barlak-Omland-Stammeier猜想。
A trace pairing and Elliott invariant for groupoid homology
本研究定义了平展群胚的Crainic-Moerdijk同调与底空间不变测度间的配对，无需底空间完全不连通，基于此提出群胚Elliott不变量，证明其与多类群胚C*-代数的对应不变量一致，细化了Matui HK猜想。
Partial actions of free groups and groupoid homology
研究自由群的偏作用群oid的同调计算及其代数的全局维数特性。
Groupoid Homology and Classifying-Space Homology Are Not Isomorphic
本文通过康托尔集上的单位群胚，证明了Matui型群胚同调与分类空间奇异同调在0阶上不同构，前者为可数群而后者基数为2^ℵ₀。
Dynamic asymptotic dimension and Matui's HK conjecture
本文证明主充足群胚的同调群在高于其动态渐近维数的维度上为零，从而将有限动态渐近维数群胚的 C*-代数 K-理论归结为群胚同调，并由此验证了动态渐近维数至多为二的主充足群胚满足 Matui 的 HK 猜想。
Cup and cap products for cohomology and homology groups of ample groupoids
本文研究丰富群胚上同调与同调中的杯积和帽积，给出清晰定义，并应用帽积分析群胚C*-代数自同构的渐近内性，首次显式计算平铺空间上同调中的杯积。
Homology and K-theory for self-similar actions of groups and groupoids
本文为自相似群作用构造长正合序列，计算其群胚同调和C*-代数K-理论，并证明包含Grigorchuk群的Röver单群有理无环但Schur乘子非平凡。
Chern character for torsion-free ample groupoids
本文构建了从 Baum-Connes 组合映射域到 Crainic-Moerdijk 群oids 带有有理系数的同调群的 Chern 符号映射，验证了 Matui 的 HK 猜想变体，并应用于超几何动力系统、拓扑满群同调等。
Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebras of various discrete groups. The examples we consider are the infinite dihedral group Z \times_σ $Z_2$ and the semidirect product group $Z \times_σ Z$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.
Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
本文受Higman-Thompson群同调研究启发，通过构造紧开双截面的小置换范畴，建立了丰富群胚、拓扑全群、代数K-理论谱与无穷环空间间的联系，并应用于计算同调不变量、证明消没及无环性结果，最终证明了Matui的AH猜想。
It is proven that Matui's AH~conjecture is true for Katsura--Exel--Pardo groupoids $\mathcal{G}_{A,B}$ associated to integral matrices $A$ and $B$. This conjecture relates the topological full group of an ample groupoid with the homology groups of the groupoid. We also give a criterion under which the topological full group $[[\mathcal{G}_{A,B}]]$ is finitely generated.
We compute the homology groups of transformation groupoids associated with odometers and show that certain infinite dihedral group odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C^*_r(G)$. We also show that transformation grupoids of odometers satisfy the AH conjecture.
We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the $K$-theory of the associated $C^*$-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the $C^*$-algebras.
Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal
本文将群胚同调与 K-理论关系应用于 Smale 空间，证明具有全不交稳定集的 Smale 空间相关谱序列在第二页展示 Putnam 稳定同调群，且该同调同构于不稳定等价关系的群胚同调。
Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids
本文建立了 ample 群体摩尔同调的广义系数定理和梅耶-维托里斯长正合列，通过链级证明提供了有效计算工具，强调系数群离散性的重要性。
Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
本文通过引入空间方向分解，将Smale空间étale群胚的同调不变量研究推广到超越稳定不连通情形，证明Putnam同调与Crainic-Moerdijk群胚同调同构，并给出不稳定C*-代数K群的谱序列计算方法。
Homology and twisted C*-algebras for self-similar actions and Zappa-Szép products
本文研究自相似作用对应的小范畴Zappa-Szép积的范畴同调，证明其可由双复形谱序列计算，并据此定义扭曲C*-代数并证明其基本结构性质。
Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid ${\mathcal G}(G,E)$ and its $C^*$-algebra $C^*(G,E)$. We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the $K$-theory of $C^*(G,E)$ and the homology of ${\mathcal G}(G,E)$ in certain cases and illustrate with an example.
Homology of Steinberg algebras
本文研究了ample群胚Steinberg代数的同调不变量，计算了特定群胚的Hochschild与循环同调，并利用Dennis迹建立了K理论到群胚同调的映射，最后针对Exel-Pardo群胚给出了具体计算。
Rigidity, Generators and Homology of Interval Exchange Groups
本文通过构造部分变换群胚将区间交换群实现为拓扑全群，证明其群胚同调与子群同调的关系及C*-代数可分类性，并给出三类关键情形的同调计算与显式生成元。
Two conjectures about homology groups, K-groups and topological full groups of minimal etale groupoids on Cantor sets are formulated. We verify these conjectures for many examples of etale groupoids including products of etale groupoids arising from one-sided shifts of finite type. Furthermore, we completely determine when these product groupoids are mutually isomorphic. Also, the abelianization of their topological full groups are computed. They are viewed as generalizations of the higher dimensional Thompson groups.
Given a self-similar groupoid action $(G,E)$ on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs $\mathcal G(G,E)$. We study the analogue of the Higman-Thompson group associated to $(G,E)$ using $G$-tables and relate it to the topological full group of $\mathcal G(G,E)$, which is isomorphic to a subgroup of unitaries in the algebra $C^*(G,E)$. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for $\mathcal G(G,E)$ in some particular cases.
Positivity of Matui's HK Conjecture for AF Groupoids
本文推广了 Matui HK 猜想在 AF 群胚上的应用，给出从同调群到 K-理论的显式序同构公式，并借此刻画 Deaconu-Renault 群胚 C*-代数的 AF 可嵌入性。
Cohomology of ample groupoids
该研究为充裕广群引入基于平坦消解的上链复形，证明其与已有连续上闭上同调一致且在Morita等价下不变，推导了斜积上同调的正合序列，给出了AF-广群等的上同调计算方法。
This is a survey of the recent development of the study of topological full groups of etale groupoids on the Cantor set. Etale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Z-actions, minimal Z^N-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so…
We show that Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.
Motivated by the search for new examples of "noncommutative manifolds", we study the noncommutative geometry of the group C*-algebras of various discrete groups. The examples we consier are the infinite dihedral group ${\bf Z} \times_σ {\bf Z}_2$ and the semidirect product ${\bf Z} \times_σ {\bf Z}$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.
HK and GL
研究多Z群在Cantor集上自由作用的变换群胚的HK猜想和间隙标记问题，通过比较群胚同调与上同调的长正合序列和Pimsner-Voiculescu正合序列，引入上同调比较映射，对Hirsch长度小的群证明或部分证明HK猜想和间隙标记。
AF Embeddability of the C*-Algebra of a Deaconu-Renault Groupoid
本文研究了 Deaconu-Renault 群胚的 C*-代数的 AF 可嵌入性，通过给出满射局部同胚的判定条件并分析同调群与 K-理论同构，推广了图与交换 C*-代数交叉积的相关定理。
Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology
本文研究了 ample 群oid 的同调，通过神经的紧支撑 Moore 复形，建立了群oid 同调的通用系数短正合序列，并推导了 Mayer-Vietoris 长正合序列。
The Hochschild homology and cohomology group can be described in terms of the homology and cohomology of the classifying space of the groupoid of the adjoint action of the group under the suitable assumption of the finiteness of the supports of cohomology groups. The difference between homology and cohomology leads to a correction of the results in the book by D. J. Benson (1995),(1991).
Maximal subgroups of topological full groups of shifts of finite type
该研究探讨有限型移位拓扑全群的极大子群，证明素子群胚的三分性，给出其拓扑全群正规化子为极大的判据，扩展Thompson群V相关结果并得到含交换子群的子群的极大子群数量结论。
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