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 Pabell´on 1 of Ciudad Universitaria courtesy of Adri´an Sarchese and Edgar Bringas. Back cover photo of Professor Luis Santal´ois 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 : Buenos Aires, Argentina) Topics in noncommutative geometry : Third Luis Santal´o Winter School-CIMPA Research School on Topics in Noncommutative Geometry, July 26–August 6, 2010, Universidad de Buenos Aires, Buenos Aires, Argentina / Guillermo Corti˜nas, editor. page cm. — (Clay mathematics proceedings ; volume 16) Includes bibliographical references. ISBN 978-0-8218-6864-5 (alk. paper) 1. Commutative algebra—Congresses. I. Corti˜nas, Guillermo, editor of compilation. II. Title. QA251.3.L85 2012 512.55—dc23 2012031426 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the Clay Mathematics Institute. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. The Clay Mathematics Institute retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://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. Sanchez-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, organized by the Mathematics Department and the Santal´o Mathematical Research 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 Santal´o. 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 (a specific 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 the quantization problem in physics. Then they review the basics on star products, the main results on deformation quantization and the notion of Morita equivalence of associative algebras. After this introductory material, they present a description 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- tive calculus defines classical algebraic structures arising from the usual calculus on 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, those noncommutative geometry structures are equivalent to the classical 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 projections in the algebra of matrices over a deformed algebra. It turns out that all the data entering into this problem (namely, a deformed algebra, a trace on it, and projections in it) can be classified using formal Poisson structures on the manifold. The algebraic index theorem implies the celebrated index theorem of Atiyah-Singer and its various generalizations. Operad theory, mentioned before as related to Kontsevich’s formality theorem, is the subject of Eduardo Hoefel’s paper. It follows from general theory that the Fulton-MacPherson operad Fn and the little discs operad Dn are equivalent. Hoefel 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 equivalence of these constructions. The author shows how these examples lead nat- urally to the concept of a strict 2-category and to the unifying approach of a functor from a group to strict 2-categories whose objects are 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)toanytwosepara- 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- ing the Pimsner-Voiculescu sequence for crossed products with Z (used for example 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∗- G algebra Cr(G) of a group G is the G-equivariant K-homology KK∗ (EG, C)of the classifying space for proper actions.