# Index Theory, Coarse Geometry, and Topology of Manifolds

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*John Roe*

A co-publication of the AMS and CBMS

The Atiyah-Singer index theorem is one of the most
powerful tools for relating geometry, analysis, and topology. In its
original form, however, it applies only to compact manifolds. This
book describes a version of index theory which works for noncompact
spaces with appropriate control, such as complete Riemannian manifolds.
The relevant “control” is provided by the large scale
geometry of the space, and basic notions of large scale geometry
are developed.

Index theory for the signature operator is related to
geometric topology via surgery theory. And, paralleling the
analytic development, “controlled” surgery theories for
noncompact spaces have been developed by topologists. This book explores
the connections between these theories, producing a natural
transformation from surgery to “analytic surgery”.

The analytic foundations of the work come from the theory
of \(C^*\)-algebras, and the properties of
the \(C^*\)-algebra of a coarse space are developed in
detail.

The book is based on lectures presented at a conference held
in Boulder, Colorado, in August 1995 and includes the author's
detailed notes and descriptions of some constructions that were
finalized after the lectures were delivered.

Also available from the AMS by John Roe is Lectures on Coarse
Geometry.

#### Reviews & Endorsements

Highly recommended for anyone interested in the relationship between index theory and the topology of manifolds.

-- Mathematical Reviews

A clear introduction to a subject which is obviously quite complicated … for those interested in acquainting themselves with the \(C^*\)-algebraic index theory and its applications, the entire book should be required reading.

-- Bulletin of the LMS

#### Table of Contents

# Table of Contents

## Index Theory, Coarse Geometry, and Topology of Manifolds

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Index Theory 112 free
- Chapter 2. Coarse Geometry 920
- Chapter 3. C*-Algebras 1728
- Chapter 4. An example of a higher index theorem 2738
- Chapter 5. Assembly 3546
- Chapter 6. Surgery 4556
- Chapter 7. Mapping surgery to analysis 5768
- Chapter 8. The coarse Baum-Connes conjecture 6980
- Chapter 9. Methods of computation 7788
- Chapter 10. Coarse structures and boundaries 8798
- References 93104
- Index 99110
- Back Cover Back Cover1113