Long exact sequences of homology groups of étale groupoids

Hiroki Matui

doi:10.3934/dcds.2022095

Article Contents

2022, Volume 42, Issue 11: 5239-5271. Doi: 10.3934/dcds.2022095

This issue Previous Article The dimensional estimates of exponential growth solutions to uniformly elliptic equations of non-divergence form Next Article Non-synchronized solutions to nonlinear elliptic Schrödinger systems on a closed Riemannian manifold

Long exact sequences of homology groups of étale groupoids

Hiroki Matui^,

Graduate School of Science, Chiba University, Japan

Corresponding author: Hiroki Matui
Corresponding author: Hiroki Matui

Received: November 2021

Revised: May 2022

Early access: July 20, 2022

Published online: July 20, 2022

This work was supported by JSPS KAKENHI Grant Number JP18K03321.

Abstract / Introduction Full Text(HTML) Figure(9) Related Papers Cited by

Abstract

When a pair of étale groupoids $ \mathcal{G} $ and $ \mathcal{G}' $ on totally disconnected spaces are related in some way, we discuss the difference of their homology groups. More specifically, we treat two basic situations. In the subgroupoid situation, $ \mathcal{G}' $ is assumed to be an open regular subgroupoid of $ \mathcal{G} $. In the factor groupoid situation, we assume that $ \mathcal{G}' $ is a quotient of $ \mathcal{G} $ and the factor map $ \mathcal{G}\to\mathcal{G}' $ is proper and regular. For each, we show that there exists a long exact sequence of homology groups. We present examples which arise from SFT groupoids and hyperplane groupoids.

Keywords:

Mathematics Subject Classification: Primary: 37B10, 37B05; Secondary: 37B52.

Citation:

Full Text(HTML)

Figure 1. the orientation of $ P_j $

Download: Full-size image PowerPoint slide

Figure 2. $ 2\theta $ translation modulo $ \mathbb{Z} $

Download: Full-size image PowerPoint slide

Figure 3. $ (2\theta{-}1)e_3 $ translation of triangle

Download: Full-size image PowerPoint slide

Figure 4. 'difference' of two triangles

Download: Full-size image PowerPoint slide

Figure 5. 'difference' of two triangles

Download: Full-size image PowerPoint slide

Figure 6. a puzzle

Download: Full-size image PowerPoint slide

Figure 7. the orientation of $ P_j $

Download: Full-size image PowerPoint slide

Figure 8. 'difference' of triangles

Download: Full-size image PowerPoint slide

Figure 9. 'difference' of triangles

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^$-algebras, Ergodic Theory Dynam. Systems*, 18 (1998), 509-537. doi: 10.1017/S0143385798100457.
[2]	C. Bönicke, C. Dell'Aiera, J. Gabe and R. Willett, Dynamic asymptotic dimension and Matui's HK conjecture, preprint, arXiv: 2104.05885.
[3]	M. Boyle, B. Kitchens and B. Marcus, A note on minimal covers for sofic systems, Proc. Amer. Math. Soc., 95 (1985), 403-411. doi: 10.1090/S0002-9939-1985-0806078-7.
[4]	M. Crainic and I. Moerdijk, A homology theory for étale groupoids, J. Reine Angew. Math., 521 (2000), 25-46. doi: 10.1515/crll.2000.029.
[5]	C. Farsi, A. Kumjian, D. Pask and A. Sims, Ample groupoids: Equivalence, homology, and Matui's HK conjecture, Münster J. Math., 12 (2019), 411-451. doi: 10.17879/53149724091.
[6]	A. H. Forrest, J. R. Hunton and J. Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys., 226 (2002), 289-322. doi: 10.1007/s002200200594.
[7]	A. Forrest, J. Hunton and J. Kellendonk, Topological invariants for projection method patterns, Mem. Amer. Math. Soc., 159 2002. doi: 10.1090/memo/0758.
[8]	T. Giordano, I. F. Putnam and C. F. Skau, $K$-theory and asymptotic index for certain almost one-to-one factors, Math. Scand., 89 (2001), 297-319. doi: 10.7146/math.scand.a-14343.
[9]	P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-0346-0189-4.
[10]	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, 2000,177–206.
[11]	H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc., 104 (2012), 27-56. doi: 10.1112/plms/pdr029.
[12]	H. Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84. doi: 10.1515/crelle-2013-0041.
[13]	H. Matui, Étale groupoids arising from products of shifts of finite type, Adv. Math., 303 (2016), 502-548. doi: 10.1016/j.aim.2016.08.023.
[14]	E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182. doi: 10.1090/S0002-9947-1951-0042109-4.
[15]	E. Ortega, The homology of the Katsura-Exel-Pardo groupoid, J. Noncommut. Geom., 14 (2020), 913-935. doi: 10.4171/JNCG/382.
[16]	V. Proietti and M. Yamashita, Homology and $K$-theory of dynamical systems. I. torsion-free ample groupoids, Ergodic Theory Dynam. Systems, 42 (2022), 2630–2660. arXiv: 2006.08028. doi: 10.1017/etds.2021.50.
[17]	V. Proietti and M. Yamashita, Homology and $K$-theory of dynamical systems. II. smale spaces with totally disconnected transversal, to appear in J. Noncommut. Geom., arXiv: 2104.10938.
[18]	I. F. Putnam, An excision theorem for the $K$-theory of $C^$-algebras, J. Operator Theory*, 38 (1997), 151-171.
[19]	I. F. Putnam, On the $K$-theory of $C^$-algebras of principal groupoids, Rocky Mountain J. Math.*, 28 (1998), 1483-1518. doi: 10.1216/rmjm/1181071727.
[20]	I. F. Putnam, Functoriality of the $C^$-algebras associated with hyperbolic dynamical systems, J. London Math. Soc.*, 62 (2000), 873-884. doi: 10.1112/S002461070000140X.
[21]	I. F. Putnam, Non-commutative methods for the $K$-theory of $C^$-algebras of aperiodic patterns from cut-and-project systems, Comm. Math. Phys., 294* (2010), 703-729. doi: 10.1007/s00220-009-0968-0.
[22]	I. F. Putnam, A Homology Theory for Smale Spaces, Mem. Amer. Math. Soc., 232 2014. doi: 10.1090/memo/1094.
[23]	I. F. Putnam, An excision theorem for the $K$-theory of $C^$-algebras, with applications to groupoid $C^$-algebras, Münster J. Math., 14 (2021), 349-402. doi: 10.17879/06089647453.
[24]	I. Putnam, K. Schmidt and C. Skau, $C^$-algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory*, 16 (1986), 99-126.
[25]	J. Renault, A Groupoid Approach to $C^ $-Algebras*, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin, 1980.
[26]	J. Renault, Cartan subalgebras in $C^$-algebras, Irish Math. Soc. Bull.*, 61 (2008), 29-63.
[27]	J. Rodríguez-López and S. Romaguera, The relationship between the Vietoris topology and the Hausdorff quasi-uniformity, Topology Appl., 124 (2002), 451-464. doi: 10.1016/S0166-8641(01)00252-8.
[28]	E. Scarparo, Homology of odometers, Ergodic Theory Dynam. Systems, 40 (2020), 2541-2551. doi: 10.1017/etds.2019.13.
[29]	I. Yi, Homology and Matui's HK conjecture for groupoids on one-dimensional solenoids, Bull. Aust. Math. Soc., 101 (2020), 105-117. doi: 10.1017/S0004972719000522.