Topics in Noncommutative Geometry Clay Mathematics Proceedings Volume 16 Topics in Noncommutative Geometry Third Luis Santaló Winter School - CIMPA Research School Topics in Noncommutative Geometry Universidad de Buenos Aires Buenos Aires, Argentina July 26–August 6, 2010 Guillermo Cortiñas Editor American Mathematical Society Clay Mathematics Institute 2010 Mathematics Subject Classification. Primary 14A22, 18D50, 19K35, 19L47, 19L50, 20F10, 46L55, 53D55, 58B34, 81R60. Cover photo of the front of Pabello´n 1 of Ciudad Universitaria courtesy of Adria´n Sarchese and Edgar Bringas. Back cover photo of ProfessorLuis Santalo´ is courtesy of his family. Library of Congress Cataloging-in-Publication Data Luis Santal´o Winter School-CIMPA Research School on Topics in Noncommutative Geometry (2010: BuenosAires,Argentina) Topics in noncommutative geometry : Third Luis Santalo´ Winter School-CIMPA Research SchoolonTopicsinNoncommutativeGeometry,July26–August6,2010,UniversidaddeBuenos Aires,BuenosAires,Argentina/GuillermoCortin˜as,editor. pagecm. —(Claymathematicsproceedings;volume16) Includesbibliographicalreferences. ISBN978-0-8218-6864-5(alk.paper) 1. Commutative algebra—Congresses. I. Cortin˜as, Guillermo, editor of compilation. II.Title. QA251.3.L85 2012 512(cid:2).55—dc23 2012031426 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 171615141312 Contents Preface vii Classifying Morita Equivalent Star Products 1 Henrique Bursztyn and Stefan Waldmann Noncommutative Calculus and Operads 19 Boris Tsygan Some Elementary Operadic Homotopy Equivalences 67 Eduardo Hoefel Actions of Higher Categories on C*-Algebras 75 Ralf Meyer Examples and Applications of Noncommutative Geometry and K-Theory 93 Jonathan Rosenberg Rational Equivariant K-Homology of Low Dimensional Groups 131 J.-F. Lafont, I. J. Ortiz and R. J. Sa´nchez-Garc´ıa Automata Groups 165 Andrzej Zuk Spectral Triples and KK-Theory: A Survey 197 Bram Mesland Deformations of the Canonical Spectral Triples 213 R. Trinchero Twisted Bundles and Twisted K-Theory 223 Max Karoubi A Guided Tour Through the Garden of Noncommutative Motives 259 Gonc¸alo Tabuada v Preface This volume contains the proceedings of the third Luis Santal´o Winter School, organizedbytheMathematicsDepartmentandtheSantalo´MathematicalResearch Institute of the School of Exact and Natural Sciences of the University of Buenos Aires (FCEN). This series of schools is named after the geometer Luis Santalo´. Born in Spain, the celebrated founder of Integral Geometry was a Professor in our department where he carried out most of his distinguished professional career. This edition of the Santal´o School took place in the FCEN from July 26 to August 6 of 2010. On this occasion the school was devoted to Noncommutative Geometry, and was supported by several institutions; the Clay Mathematics Insti- tute was one of its main sponsors. The topics of the school and the contents of this volume concern Noncommu- tative Geometry in a broad sense: it encompasses the various mathematical and physical theories that incorporate geometric ideas to study noncommutative phe- nomena. One of those theories is that of deformation quantization. A main result in this area is Kontsevich’s formality theorem. It implies that a Poisson structure on a manifold can always be formally quantized. More precisely it shows that there is an isomorphism (although not canonical) between the moduli space of formal deformations of Poisson structures on a manifold and the moduli space of star products on the manifold. The question of understanding Morita equivalence of star products under (aspecific choice of) this isomorphism was solved by Henrique Bursztyn and Stefan Waldmann; their article in the present volume gives a survey of their work. They start by discussing how deformation quantization arises from thequantizationprobleminphysics. Thentheyreviewthebasicsonstarproducts, the main results on deformationquantization and the notion of Moritaequivalence ofassociativealgebras. Afterthisintroductorymaterial, theypresent adescription of Morita equivalence for star products as orbits of a suitable group action on the one hand, and the B-field action on (formal) Poisson structures on the other. Finally, they arrive at their main results on the classification of Morita equivalent star products. The next article, by Boris Tsygan, reviews Tamarkin’s proof of Kontsevich’s formality theorem using the theory of operads. It also explains applications of the formality theorem to noncommutative calculus and index theory. Noncommuta- tivecalculusdefinesclassicalalgebraicstructuresarisingfromtheusualcalculuson manifolds in terms of the algebra of functions on this manifold, in a way that is vii viii PREFACE valid for any associative algebra, commutative or not. It turns out that noncom- mutative analogs of the basic spaces arising in calculus are well-known complexes from homological algebra. These complexes turn out to carry a very rich algebraic structure, similar to the one carried by their classical counterparts. It follows from Kontsevich’s formality theorem that when the algebra in question is the algebra of functions,thosenoncommutativegeometrystructuresareequivalenttotheclassical ones. Another consequence is the algebraic index theorem for deformation quanti- zations. This is a statement about a trace of a compactly supported difference of projectionsinthealgebraofmatricesoveradeformedalgebra. Itturnsoutthatall the dataentering into this problem (namely, a deformed algebra, atrace on it, and projections in it) can be classified using formal Poisson structures on the manifold. ThealgebraicindextheoremimpliesthecelebratedindextheoremofAtiyah-Singer and its various generalizations. Operadtheory, mentionedbeforeasrelatedtoKontsevich’sformalitytheorem, is the subject of Eduardo Hoefel’s paper. It follows from general theory that the Fulton-MacPhersonoperadF andthelittlediscsoperadD areequivalent. Hoefel n n gives an elementary proof of this fact by exhibiting a rather explicit homotopy equivalence between them. Many important examples in noncommutative geometry appear as crossed products. Ralf Meyer’s lecture notes deal with several crossed-products of C∗- algebras, (e.g. crossed-products by actions of locally compact groups, twisted crossed-products, crossed products by C∗correspondences) and discuss notions of equivalenceoftheseconstructions. Theauthorshowshowtheseexamplesleadnat- urallytotheconceptofastrict2-categoryandtotheunifyingapproachofafunctor froma group tostrict 2-categories whose objectsare C∗-algebras. In addition, this approach enables the author to define in all generality the crossed product of a C∗-algebra by a group acting by Morita-Rieffel bimodules. Jonathan Rosenberg’s article is concerned with two related but distinct top- ics: noncommutative tori and Kasparov’s KK-theory. Noncommutative tori are certain crossed products of the algebra of continuous functions on the unit circle by an action of Z. They turn out to be noncommutative deformations of the al- gebra of continuous functions on the 2-torus. The article reviews the classification of noncommutative tori up to Morita equivalence, and of bundles (i.e. projec- tive modules) over them, as well as some applications of noncommutative tori to number theory and physics. Kasparov’s K-theory is one of the main homological invariants in C∗-algebra theory. It can be presented in several equivalent manners; the article reviews Kasparov’s original definition in terms of Kasparov bimodules (or generalized elliptic pseudodifferential operators), Cuntz’s picture in terms of quasi-homomorphisms, and Higson’s description of KK as a universal homology theory for separable C∗-algebras. It assigns groups KK∗(A,B) to any two separa- ble C∗-algebras A and B; usual operator K-theory and K-homology are recovered by setting A (respectively B) equal to the complex numbers. The article also considers the equivariant version of KK for C∗-algebras equipped with a group action, and considers its applications to the K-theory of crossed products, includ- ingthePimsner-VoiculescusequenceforcrossedproductswithZ(usedforexample to compute the K-theory of noncommutative tori), Connes’ Thom isomorphism PREFACE ix for crossed products with R and the Baum-Connes conjecture for crossed products with general locally compact groups. The Baum-Connes conjecture predicts that the K-theory of the reduced C∗- algebra C (G) of a group G is the G-equivariant K-homology KKG(EG,C) of r ∗ the classifying space for proper actions. The conjecture is known to hold in many cases. The point of the conjecture is that the K-homology is in principle easier to compute than the K-theory of C (G), but concrete computations are often very r hard, especially if the group contains torsion. The article by Jean-Fran¸cois Lafont, Ivonne Ortiz and Rub´en Sa´nchez-Garc´ıa is concerned with computing the groups KKG(EG,C) ⊗Q in the case where G admits a 3-dimensional manifold model ∗ M3 for EG. In this case the authors provide an explicit formula in terms of the combinatorics of the model M3. As noted above, groups play a key role in noncommutative geometry; group theoryisthereforeanimportant tool inthearea. Andrzej Zuk’s articlepresents an introduction to the theory of groups generated by finite automata. This class con- tainsseveralremarkablecountablegroups,whichprovidesolutionstolongstanding questionsinthefield. Forinstance,Aleshin’sautomatagivesagroupwhichisinfin- tely generated yet torsion; this answers affirmatively a question raised by Burnside in 1902 of whether such groups could exist. Aleshin’s group is also an example of a group which is not of polynomial growth yet it is of subexponential growth. Other remarkable examples of automata groups are also presented, including a group without uniform exponential growth, and exotic amenable groups. In Connes’ theory of noncommutative geometry, spectral triples play the role of noncommutative Riemannian manifolds; they are also the source of elements in Kasparov’s KK-theory. Bram Mesland’s article deals with the construction of a category of spectral triples that is compatible with the Kasparov product in KK- theory. The theory described shows that by introducing a notion of smoothness on unboundedKK-cycles,theKasparovproductofsuchcyclescanbedefineddirectly, by an algebraic formula. This allows one to view such cycles as morphisms in a category whose objects are spectral triples. ThearticlebyRobertoTrincheroisalsoconcernedwithspectraltriples. Itpro- vides a link betweenConnes’ noncommutative geometry and quantum field theory. It displays, in detail, a toy model that aims to shed some light in the dimensional renormalization framework of Quantum Field Theory, by comparison with the re- sults obtained under the paradigm of such theory, where a Grasmannian algebra is the starting point. MaxKaroubi’sarticleconcernstwistedK-theory,atheoryofveryactivecurrent research, which was originally defined by Karoubi and Donovan in the late 1960s. Theapproachpresentedhereisbasedonthenotionoftwisted vector bundles. Such bundles may be interpreted as modules over suitable algebra bundles. Roughly speaking, twisted K-theory appears as the Grothendieck group of the category of twisted vector bundles. Thus a geometric description of the theory is obtained. The usual operations on vector bundles are extended to twisted vector bundles. The article also contains a section on cup-products, where it is shown that the