Higher index theory / Rufus Willett, Guoliang Yu.

Format:

Book

Author/Creator:

Willett, Rufus, 1983- author.

Yu, Guoliang, 1963 August 11- author.

Series:

Cambridge studies in advanced mathematics ; 189.

Cambridge studies in advanced mathematics ; 189

Language:

English

Subjects (All):

Index theory (Mathematics).

C*-algebras.

K-theory.

Physical Description:

xi, 581 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2020.

Summary:

"Index theory studies the solutions to differential equations on geometric spaces, their relation to the underlying geometry and topology, and applications to physics. If the space of solutions is infinite dimensional, it becomes necessary to generalise the classical Fredholm index using tools from the Ktheory of operator algebras. This leads to higher index theory, a rapidly developing subject with connections to noncommutative geometry, large-scale geometry, manifold topology and geometry, and operator algebras. Aimed at geometers, topologists and operator algebraists, this book takes a friendly and concrete approach to this exciting theory, focusing on the main conjectures in the area and their applications outside of it. A well-balanced combination of detailed introductory material (with exercises), cutting-edge developments and references to the wider literature make this a valuable guide to this active area for graduate students and experts alike"-- Provided by publisher

Contents:

Part I. Background:

C*-algebras

K-theory for C*-algebras

Motivation: positive scalar curvature on tori

Part II. Roe algebras, Localisation algebras, and assembly:

Geometric modules

Roe algebras

Localisation algebras and K-homology

Assembly maps and the Baum-Connes conjecture

Part III. Differential operators:

Elliptic operators and K-homology

Products and Poincaré duality

Applications to algebra, geometry, and topology

Part IV. Higher index theory and assembly:

Almost constant bundles

Higher index theory for coarsely embeddable spaces

Counterexamples

Appendix A. Topological spaces, group actions, and coarse geometry

Appendix B. Categories of topological spaces and homology theories

Appendix C. Unitary representations

Appendix D. Unbounded operators

Appendix E. Gradings.

Notes:

Includes bibliographical references and indexes.

Other Format:

Online version: Willett, Rufus, 1983- Higher index theory.

ISBN:

9781108491068

1108491065

OCLC:

1141923991