Higher Index Theory

Friendly introduction to higher index theory, a growing subject at the intersection of geometry, topology and operator algebras.

Rufus Willett (Author), Guoliang Yu (Author)

9781108491068, Cambridge University Press

Hardback, published 2 July 2020

592 pages, 1 b/w illus. 150 exercises
23.4 x 15.8 x 3.6 cm, 0.94 kg

'This book is an exceptional blend of clear, concise, delightfully written exposition and thorough scholarship. The book also has much to offer more experienced researchers.' Peter Haskell, European Mathematical Society

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 K-theory 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.

Introduction
Part I. Background: 1. C*-algebras
2. K-theory for C*-algebras
3. Motivation: positive scalar curvature on tori
Part II. Roe Algebras, Localisation Algebras, and Assembly: 4. Geometric modules
5. Roe algebras
6. Localisation algebras and K-homology
7. Assembly maps and the Baum–Connes conjecture
Part III. Differential Operators: 8. Elliptic operators and K-homology
9. Products and Poincaré duality
10. Applications to algebra, geometry, and topology
Part IV. Higher Index Theory and Assembly: 11. Almost constant bundles
12. Higher index theory for coarsely embeddable spaces
13. 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
References
Index of symbols
Subject index.

Subject Areas: Algebraic topology [PBPD], Calculus & mathematical analysis [PBK]