Proper Group Actions and the Baum-Connes Conjecture
Proper Group Actions and the Baum-Connes Conjecture

Author: Guido Mislin

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 138

ISBN-13: 3034880898

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A concise introduction to the techniques used to prove the Baum-Connes conjecture. The Baum-Connes conjecture predicts that the K-homology of the reduced C^*-algebra of a group can be computed as the equivariant K-homology of the classifying space for proper actions. The approach is expository, but it contains proofs of many basic results on topological K-homology and the K-theory of C^*-algebras. It features a detailed introduction to Bredon homology for infinite groups, with applications to K-homology. It also contains a detailed discussion of naturality questions concerning the assembly map, a topic not well documented in the literature. The book is aimed at advanced graduate students and researchers in the area, leading to current research problems.

Introduction to the Baum-Connes Conjecture
Introduction to the Baum-Connes Conjecture

Author: Alain Valette

Publisher: Springer Science & Business Media

Published: 2002-04-01

Total Pages: 120

ISBN-13: 9783764367060

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The Baum-Connes conjecture is part of A. Connes' non-commutative geometry programme. It can be viewed as a conjectural generalisation of the Atiyah-Singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group "gamma"). Like the Atiyah-Singer theorem, the Baum-Connes conjecture states that a purely topological object coincides with a purely analytical one. For a given group "gamma", the topological object is the equivariant K-homology of the classifying space for proper actions of "gamma", while the analytical object is the K-theory of the C*-algebra associated with "gamma" in its regular representation. The Baum-Connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra. It has also strong connections with geometric group theory, as the proof of the conjecture for a given group "gamma" usually depends heavily on geometric properties of "gamma". This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. It presents, for the first time in book form, an introduction to the Baum-Connes conjecture. It starts by defining carefully the objects in both sides of the conjecture, then the assembly map which connects them. Thereafter it illustrates the main tool to attack the conjecture (Kasparov's theory), and it concludes with a rough sketch of V. Lafforgue's proof of the conjecture for co-compact lattices in in Spn1, SL(3R), and SL(3C).

Introduction to the Baum-Connes Conjecture
Introduction to the Baum-Connes Conjecture

Author: Alain Valette

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 111

ISBN-13: 3034881878

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The Baum-Connes conjecture is part of A. Connes' non-commutative geometry programme. It can be viewed as a conjectural generalisation of the Atiyah-Singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group "gamma"). Like the Atiyah-Singer theorem, the Baum-Connes conjecture states that a purely topological object coincides with a purely analytical one. For a given group "gamma", the topological object is the equivariant K-homology of the classifying space for proper actions of "gamma", while the analytical object is the K-theory of the C*-algebra associated with "gamma" in its regular representation. The Baum-Connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra. It has also strong connections with geometric group theory, as the proof of the conjecture for a given group "gamma" usually depends heavily on geometric properties of "gamma". This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. It presents, for the first time in book form, an introduction to the Baum-Connes conjecture. It starts by defining carefully the objects in both sides of the conjecture, then the assembly map which connects them. Thereafter it illustrates the main tool to attack the conjecture (Kasparov's theory), and it concludes with a rough sketch of V. Lafforgue's proof of the conjecture for co-compact lattices in in Spn1, SL(3R), and SL(3C).

Topics in Algebraic and Topological K-Theory
Topics in Algebraic and Topological K-Theory

Author: Paul Frank Baum

Publisher: Springer

Published: 2010-10-28

Total Pages: 322

ISBN-13: 3642157084

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This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Infinite Groups: Geometric, Combinatorial and Dynamical Aspects
Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Author: Laurent Bartholdi

Publisher: Springer Science & Business Media

Published: 2006-03-28

Total Pages: 419

ISBN-13: 3764374470

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This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.

K-Theory for Group C*-Algebras and Semigroup C*-Algebras
K-Theory for Group C*-Algebras and Semigroup C*-Algebras

Author: Joachim Cuntz

Publisher: Birkhäuser

Published: 2017-10-24

Total Pages: 325

ISBN-13: 3319599151

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This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.

Handbook of Homotopy Theory
Handbook of Homotopy Theory

Author: Haynes Miller

Publisher: CRC Press

Published: 2020-01-23

Total Pages: 982

ISBN-13: 1351251619

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The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Advances in Noncommutative Geometry
Advances in Noncommutative Geometry

Author: Ali Chamseddine

Publisher: Springer Nature

Published: 2020-01-13

Total Pages: 753

ISBN-13: 3030295974

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This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.

Geometric Group Theory
Geometric Group Theory

Author: Goulnara N. Arzhantseva

Publisher: Springer Science & Business Media

Published: 2007-09-24

Total Pages: 256

ISBN-13: 3764384123

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This volume has its origins in the Barcelona Conference in Group Theory (July 2005) and the conference "Asymptotic and Probabilistic Methods in Geometric Group Theory" held in Geneva (June 2005). Twelve peer-reviewed research articles written by experts in the field present the most recent results in abstract and geometric group theory. In particular there are two articles by A. Juhász.