Higher Index Theory

Rufus Willett is Professor of Mathematics at the University of Hawaii, Manoa. He has interdisciplinary research interests across large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras. Guoliang Yu is the Powell Chair in Mathematics and University Distinguished Professor at Texas A & M University. He was an invited speaker at the International Congress of Mathematicians in 2006, is a Fellow of the American Mathematical Society and a Simons Fellow in Mathematics. His research interests include large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras.

• 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.

'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