**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2007; 205 pp; Softcover

MSC: Primary 58;

Print ISBN: 978-0-8218-2097-1

Product Code: MMONO/235

Print ISBN: 978-0-8218-2097-1

Product Code: MMONO/235

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

**Electronic ISBN: 978-1-4704-4789-2
Product Code: MMONO/235.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

#### Supplemental Materials

# Index Theorem. 1

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*Mikio Furuta*

The Atiyah-Singer index theorem is a remarkable result that allows one to
compute the space of solutions of a linear elliptic partial differential
operator on a manifold in terms of purely topological data related to the
manifold and the symbol of the operator. First proved by Atiyah and Singer
in 1963, it marked the beginning of a completely new direction of research
in mathematics with relations to differential geometry, partial
differential equations, differential topology, K-theory, physics, and
other areas.

The author's main goal in this volume is to give a complete proof of the
index theorem. The version of the proof he chooses to present is the one
based on the localization theorem. The prerequisites include a first
course in differential geometry, some linear algebra, and some facts about
partial differential equations in Euclidean spaces.

#### Readership

Graduate students interested in index theory.

#### Reviews & Endorsements

The book is well organized... The strategy of the proof and applications are clearly laid out. ...this monograph is an important contribution to the subject.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Index Theorem. 1

- Cover Cover11
- Half title page i2
- Title Page iii4
- Contents v6
- Preface vii8
- Outline of the theory and perspective ix10
- Prelude 120
- Manifolds, vector bundles and elliptic complexes 2746
- Index and its localization 6382
- Examples of the localization of the index 93112
- Localization of eigenfunctions of the operator of Laplace type 125144
- Formulation and proof of the index theorem 157176
- Characteristic classes 177196
- Index 203222
- Back Cover Back Cover1225