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Volume 235
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Index Theorem. 1
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N Mikio Furuta
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American Mathematical Society
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Index Theorem. 1
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MATHEMATICAL
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MONOGRAPHS
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Volume 235
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Index Theorem. 1
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Mikio Furuta
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Translated by
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Kauro Ono
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W American Mathematical Society
Providence, Rhode Island
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Editorial Board
Shoshichi Kobayashi (Chair)
Masamichi Takesaki
(INDEX THEOREM 1)
by Mikio Furuta
Copyright © 1999 by Mikio Furuta
Originally published in Japanese
by Iwanami Shoten, Publishers, Tokyo, 1999
Translated from the Japanese by Kaoru Ono
2000 Mathematics Subject Classification. Primary 58-01, 58J20.
For additional information and updates on this book, visit
www.ams.org/bookpages/mmono-235
Library of Congress Cataloging-in-Publication Data
Furuta, M.
Index theorem. 1 / Mikio Furuta ; translated by Kaoru Ono.
p. cm. — (Translations of mathematical monographs ; v. 235) (Iwanami
series in modern mathematics)
Includes bibliographical references and index.
ISBN 978-0-8218-2097-1 (alk. paper)
1. Index theorems.
QA614.92.F87 2007
514'.74—dc22 2007060301
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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07
Contents
Preface vii
Outline of the Theory and Perspective ix
Chapter 1. Prelude 1
1. What Is the Index? 1
2. What Is the Atiyah-Singer Index Theorem? 11
3. 1-Dimensional Case 20
Chapter 2. Manifolds, Vector Bundles and Elliptic Complexes 27
1. Differential Forms with Compact Support and Their
Integration 27
2. Embeddings of Manifolds and Vector Bundles to Trivial
Objects 30
3. Clifford Modules and Operators of Dirac Type 34
4. Elliptic Differential Operators Appearing in Geometry
and Operators of Dirac Type 54
Chapter 3. Index and Its Localization 63
1. Definition of the Index of Operators of Dirac Type on
Closed Manifolds 63
2. Definition of the Index of Operators of Dirac Type on
Open Manifolds 65
3. The Excision Theorem and Topological Invariance of the
Index 76
4. Products of Operators of Dirac Type and Their Indices 79
5. Supersymmetric Harmonic Oscillator and the de Rham
Complex on the Euclidean Space 81
Chapter 4. Examples of the Localization of the Index 93
1. The Poincaré-Hopf Theorem and the Morse Inequality 93
2. Riemann-Roch Theorem on Riemann Surfaces 104
V
vi CONTENTS
3. Mod 2 Index of Spin Structures on Riemann Surfaces 107
4. The Case with a Group Action: Lefschetz Formula 116
Chapter 5. Localization of Eigenfunctions of the Operator of
Laplace Type 125
1. Set-up 125
2. Exponential Decay 128
3. Preliminaries for the Calculus of Variation 131
4. Calculus of Variations 138
5. Oscillation of Eigenvalues and Eigenfunctions 143
6. Modification of Operators on Ends 148
7. The Case of Closed Manifolds: Spectral Decomposition 152
Chapter 6. Formulation and Proof of the Index Theorem 157
1. Strategy for Our Formulation and Proof 157
2. Construction of the Pair on the Euclidean Space 160
3. Invariance of the Index: Proof 1 (The Index of the
Product) 164
4. Invariance of the Index: Proof 2 (Excision Theorem) 174
5. Pairs on Even Dimensional Euclidean Spaces 174
Chapter 7. Characteristic Classes 177
1. Connection and Curvature 177
2. Chern Character and Chern Classes 181
3. Localization of Chern Character 186
4. Thom Class and Thom Isomorphism 195
5. The Euler Class 199
Index 203
Preface
Linear differential equations are looked upon from two points of
view.
The first point of view is based on understanding a linear differ
ential equation as a linear mapping between two infinite-dimensional
vector spaces. Let us consider the situation when the linear map
ping gives an “almost” linear isomorphism. Here, “almost” means
“ignoring finite-dimensional vector subspaces.” In other words, the
finite-dimensional subspaces, which are ignored, are responsible for
the possibility that the two infinite-dimensional vector spaces are
not canonically isomorphic. This situation yields a linear mapping
between certain finite-dimensional vector subspaces. The object ob
tained by using this finite-dimensional approximation is called the
index.
Another point of view is based on the description of a linear dif
ferential equation using local coordinates. Then the linear differential
equation can be formally regarded as a section of a certain fiber bun
dle. The object obtained using this approach is the homotopy class of
the section and is called (the homotopy class of) the principal symbol.
Each object is invariant under continuous deformation, with ap
propriate requirements, of the linear differential equation. In this
sense, both of them are topological objects.
The two points of view are related to each other. The condition
that the principal symbol is almost invertible is a sufficient condition
for the linear differential operator to be an almost linear isomorphism
(at least on closed manifolds). The former is called ellipticity for par
tial differential equations, and the latter is called Fredholm property,
which is a notion in functional analysis. Ellipticity implies Fredholm
property.
In order to compute the index following its definition, we must
solve the linear differential equation to obtain a finite-dimensional
Vll
viii PREFACE
approximation. Among various approaches to the index, the most
basic one is given by the integer
ind P = dim Ker P — dim Coker P,
where we denote the linear mapping by P.
The Atiyah-Singer index theorem describes how to determine the
index of an elliptic differential operator by its principal symbol using
only topological tools.
As a slogan, the index theorem states the coincidence of the an
alytical index, which is the original definition, and the topological
index, which is defined using topological tools.
The index theorem as a coincidence of two numbers originated
in the work of Atiyah and Singer. Its applications to differential
topology have also been explored since then, but this area of research
is not active at this moment. In recent years, the index theorem has
become rather a daily tool used in global analysis involving non-linear
differential equations. A few decades ago, the index theorem itself was
the focal point of the formation of global analysis.
Is research on the index theorem “done with”?
The author believes that this is not the case. One piece of ev
idence is that there are still active research fields surrounding the
index theorem. Outside of such obviously related places, it seems to
the author that the study of each subject in the background of the in
dex theorem is still in progress. It is not yet clear what future theory
the index theorem uses as its first step, but surely it will be something
interesting. The contours of this future theory can be vaguely seen
when we review the development of the index theorem.
At this moment, it is important to understand the index theorem
not as a mountain chain, behind grand theories, but to digest it as
a natural element of the scenery when we follow the fundamental
properties of differential equations.
I would like to present such understanding of the index theorem
by giving proofs that are as elementary as possible. This is what I
always had in mind when writing this book. I would be happy if
readers would give opinions and criticism.
A part of this book was written during the author’s stay at the
Max-Planck-Institut fiir Mathematik. I appreciate the institute for
providing such a good working environment.
October, 1998
Mikio Furuta
Outline of the Theory and Perspective
de Rham complex and Dolbeault complex
The de Rham complex and the Dolbeault complex are two com
plexes that appear in the geometry of differentiable manifolds and the
geometry of complex manifolds, respectively.
The former, resp. the latter, has its origin in Cartan’s discovery
of the exterior derivative, resp. Cauchy’s discovery of the Cauchy-
Riemann equation.1 In the former case, its connection with geometry
was revealed when the de Rham theorem clarified its relation to Betti
numbers. In the latter case, it goes back to an older time, when
Riemann found the ancestor of the Riemann-Roch theorem, i.e., the
inequality involving the genus of the Riemann surface. However, the
former was understood as the device deriving invariants of manifolds
and the latter was understood as some properties of various holomor-
phic line bundles (or divisors). Hence they play different roles. This
is still the case now.
These two trends met in a clear form under the viewpoint of
unification in the theory of Hodge and Kodaira, which followed Weyl’s
work. Cohomology is associated with a complex and has geometric
meaning. (For clear understanding, it was also necessary to establish
the notion of sheaves and their Cech cohomology, which is equivalent
to the two kinds of complexes mentioned above.) Hodge-Kodaira
described their cohomologies in terms of solutions of the equation of
Laplace type.
Considering the alternating sum of the dimension of the coho
mology of the complex instead of the dimension of the cohomology
in each degree, we lose a lot of detailed information, but we acquire
simple properties (with respect to exact sequences). This fact is clear
•’•Manifesting the origin may depend on personal opinion. Other descriptions
in this part are also like virtual history.
IX
