Homology and 𝐾-theory of dynamical systems III. Beyond stably disconnected Smale spaces

Proietti, Valerio; Yamashita, Makoto

doi:10.1090/tran/9353

Homology and $K$-theory of dynamical systems III. Beyond stably disconnected Smale spaces
HTML articles powered by AMS MathViewer

by Valerio Proietti and Makoto Yamashita;

Trans. Amer. Math. Soc. 378 (2025), 2129-2155

DOI: https://doi.org/10.1090/tran/9353

Published electronically: December 30, 2024

HTML | PDF

Abstract:

We study homological invariants of étale groupoids arising from Smale spaces, continuing our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology groups defined by Putnam are isomorphic to the Crainic–Moerdijk groupoid homology with integer coefficients.

We also show that the $K$-groups of C$^*$-algebras of stable and unstable equivalence relations have finite rank. For unstably disconnected Smale spaces, we provide a cohomological spectral sequence whose second page shows Putnam’s (stable) homology groups, and converges to the $K$-groups of the unstable C$^*$-algebra.

References

Jared E. Anderson and Ian F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 509–537. MR 1631708, DOI 10.1017/S0143385798100457
Christian Bönicke, Clément Dell’Aiera, James Gabe, and Rufus Willett, Dynamic asymptotic dimension and Matui’s HK conjecture, Proc. Lond. Math. Soc. 126 (2023), no. 4, 1182–1253, available at https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms.12510., DOI 10.1112/plms.12510
Christian Bönicke and Valerio Proietti, A categorical approach to the Baum-Connes conjecture for étale groupoids, J. Inst. Math. Jussieu 23 (2024), no. 5, 2319–2364. MR 4821559, DOI 10.1017/S1474748023000531
Michael Brin and Garrett Stuck, Introduction to dynamical systems, Cambridge University Press, Cambridge, 2002. MR 1963683, DOI 10.1017/CBO9780511755316
Marius Crainic and Ieke Moerdijk, A homology theory for étale groupoids, J. Reine Angew. Math. 521 (2000), 25–46. MR 1752294, DOI 10.1515/crll.2000.029
Robin J. Deeley, A counterexample to the HK-conjecture that is principal, Ergodic Theory Dynam. Systems 43 (2023), no. 6, 1829–1846. MR 4583796, DOI 10.1017/etds.2022.25
Robin J. Deeley, Magnus Goffeng, and Allan Yashinski, Smale space $C^*$-algebras have nonzero projections, Proc. Amer. Math. Soc. 148 (2020), no. 4, 1625–1639. MR 4069199, DOI 10.1090/proc/14837
Robin J. Deeley, D. Brady Killough, and Michael F. Whittaker, Dynamical correspondences for Smale spaces, New York J. Math. 22 (2016), 943–988. MR 3576278
Robin J. Deeley, D. Brady Killough, and Michael F. Whittaker, Functorial properties of Putnam’s homology theory for Smale spaces, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1411–1440. MR 3519418, DOI 10.1017/etds.2014.134
Robin J. Deeley and Karen R. Strung, Nuclear dimension and classification of $\rm C^*$-algebras associated to Smale spaces, Trans. Amer. Math. Soc. 370 (2018), no. 5, 3467–3485. MR 3766855, DOI 10.1090/tran/7046
Siegfried Echterhoff and Heath Emerson, Structure and $K$-theory of crossed products by proper actions, Expo. Math. 29 (2011), no. 3, 300–344. MR 2820377, DOI 10.1016/j.exmath.2011.05.001
Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952. MR 50886, DOI 10.1515/9781400877492
Ryszard Engelking, Dimension theory, North-Holland Mathematical Library, vol. 19, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author. MR 482697
Carla Farsi, Alex Kumjian, David Pask, and Aidan Sims, Ample groupoids: equivalence, homology, and Matui’s HK conjecture, Münster J. Math. 12 (2019), no. 2, 411–451. MR 4030921, DOI 10.17879/53149724091
Zbigniew Fiedorowicz and Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57–87. MR 998125, DOI 10.1090/S0002-9947-1991-0998125-4
Roger Godement, Topologie algébrique et théorie des faisceaux, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée; Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1252. MR 345092
Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2009. Reprint of the 1999 edition [MR1711612]. MR 2840650, DOI 10.1007/978-3-0346-0189-4
Jerome Kaminker, Ian F. Putnam, and Michael F. Whittaker, K-theoretic duality for hyperbolic dynamical systems, J. Reine Angew. Math. 730 (2017), 263–299. MR 3692021, DOI 10.1515/crelle-2014-0126
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Wolfgang Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980), no. 3, 239–250. MR 561973, DOI 10.1007/BF01390047
Pierre-Yves Le Gall, Théorie de Kasparov équivariante et groupoïdes. I, $K$-Theory 16 (1999), no. 4, 361–390 (French, with English and French summaries). MR 1686846, DOI 10.1023/A:1007707525423
Kengo Matsumoto, Topological conjugacy of topological Markov shifts and Ruelle algebras, J. Operator Theory 82 (2019), no. 2, 253–284. MR 4015953, DOI 10.7900/jot
Hiroki Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc. (3) 104 (2012), no. 1, 27–56. MR 2876963, DOI 10.1112/plms/pdr029
Hiroki Matui, Étale groupoids arising from products of shifts of finite type, Adv. Math. 303 (2016), 502–548. MR 3552533, DOI 10.1016/j.aim.2016.08.023
Ralf Meyer, Homological algebra in bivariant $K$-theory and other triangulated categories. II, Tbil. Math. J. 1 (2008), 165–210. MR 2563811
Ralf Meyer and Ryszard Nest, Homological algebra in bivariant $K$-theory and other triangulated categories. I, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 236–289. MR 2681710
Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 1, 3–22. MR 873460
Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164, DOI 10.1090/surv/117
Volodymyr Nekrashevych, $C^*$-algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59–123. MR 2526786, DOI 10.1515/CRELLE.2009.035
S. Nishikawa and V. Proietti, Groups with Spanier–Whitehead duality, Ann. $K$-theory 5 (2020), no. 3, 465–500., DOI 10.2140/akt.2020.5.465
N. Christopher Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49–114. MR 1745197, DOI 10.4171/dm/75
Valerio Proietti, A note on homology for Smale spaces, Groups Geom. Dyn. 14 (2020), no. 3, 813–836. MR 4167022, DOI 10.4171/ggd/564
Valerio Proietti and Makoto Yamashita, Homology and $K$-theory of dynamical systems I. Torsion-free ample groupoids, Ergodic Theory Dynam. Systems 42 (2022), no. 8, 2630–2660. MR 4448401, DOI 10.1017/etds.2021.50
Valerio Proietti and Makoto Yamashita, Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal, J. Noncommut. Geom. 17 (2023), no. 3, 957–998. MR 4626307, DOI 10.4171/jncg/494
Valerio Proietti and Makoto Yamashita, Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology, Ergodic Theory Dynam. Systems 45 (2025), no. 1, 247–273. MR 4833663, DOI 10.1017/etds.2024.37
Ian F. Putnam, $C^*$-algebras from Smale spaces, Canad. J. Math. 48 (1996), no. 1, 175–195. MR 1382481, DOI 10.4153/CJM-1996-008-2
Ian F. Putnam, Functoriality of the $C^*$-algebras associated with hyperbolic dynamical systems, J. London Math. Soc. (2) 62 (2000), no. 3, 873–884. MR 1794291, DOI 10.1112/S002461070000140X
Ian F. Putnam, A homology theory for Smale spaces, Mem. Amer. Math. Soc. 232 (2014), no. 1094, viii+122. MR 3243636, DOI 10.1090/memo/1094
Ian F. Putnam, Lecture notes on smale spaces, 2015. Accessed: 2010-09-30.
Ian F. Putnam and Jack Spielberg, The structure of $C^*$-algebras associated with hyperbolic dynamical systems, J. Funct. Anal. 163 (1999), no. 2, 279 –299., DOI 10.1006/jfan.1998.3379
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
David Ruelle, Thermodynamic formalism, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. The mathematical structures of equilibrium statistical mechanics. MR 2129258, DOI 10.1017/CBO9780511617546
B. Saint-Donat, Techniques de descente cohomologique, Théorie des topos et cohomologie étale des schémas. tome 2., 1972. SGA4, Vbis., DOI 10.1007/BFb0061321
Eduardo Scarparo, Homology of odometers, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2541–2551. MR 4130816, DOI 10.1017/etds.2019.13
Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393, DOI 10.1007/BF02684591
Klaus Thomsen, $C^*$-algebras of homoclinic and heteroclinic structure in expansive dynamics, Mem. Amer. Math. Soc. 206 (2010), no. 970, x+122. MR 2667385, DOI 10.1090/S0065-9266-10-00581-8
Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, $K$-Theory 17 (1999), no. 3, 215–264 (French, with English and French summaries). MR 1703305, DOI 10.1023/A:1007744304422