The Atiyah-Singer Index Theorem An Introduction
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Patrick Shanahan
The Atiyah-Singer Index Theorem An Introduction
Springer-Verlag Berlin Heidelberg New York 1978
Author Patrick Shanahan Department of Mathematics Holy Cross College Worcester, MA 01610
U.S.A
AMS Subject Classifications (1970): 58G05, 58G10, 55B15, 55F40, 55F50,57E99
ISBN 3-540-08660-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08660-9 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface These notes are an expanded version of three expository lectures given at the University of Sussex in June, 1975.
They
are based primarily on the papers "The Index of Elliptic Operators I, III" of Atiyah and Singer ([2], [3]) and the paper "The Index of Elliptic Operators II" ([2]) of Atiyah and Segal which appeared in the Annals of Mathematics in 1968. It is thirteen years since Atiyah and Singer announced their index theorem in [1].
Although a number of articles dealing with
one aspect or another of the index theorem have been written since then, there are still only three major sources to which a student of the index theorem may turn for a comprehensive treatment of the sUbject: the studies edited by palais ([1]) and by cartan ([1]), and the series of three papers referred to above.
The first two
of these sources contain the original proof of the index theorem by means of cobordism and were an invaluable contribution to the literature on the subject when they appeared in 1965.
The third
gives the generalized version of the index theorem (the G-index theorem), as well as a proof of the theorem which does not depend on the results of cobordism theory.
It is an elegant example
of the art of writing clearly and succinctly about difficult mathematics. Still, because of the complexity of the sUbject, it is fair to say that none of these works is really accessible to the average mathematician who wants to learn about the index theorem and its applications, but who does not intend to attempt to become
IV
an expert on the subject.
Since the index theorem is one of the
fundamental mathematical discoveries of recent decades, it is clearly desirable that a wider segment of the mathematical community be acquainted with at least the outline of the theorem and its applications.
For this reason, an exposition falling between the
various brief surveys which have appeared and the comprehensive presentations seems called for.
It is my hope that the present notes
will at least partia
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