Curvature and Characteristic Classes
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640 Johan L. Dupont
Curvature and Characteristic Classes
Springer-Verlag Berlin Heidelberg New York 1978
Author Johan L. Dupont Matematisk Institut Ny Munkegade DK-BOOO Aarhus C/Denmark
AMS Subject Classifications (1970): 53C05, 55F40, 57D20, 58AlO, 55J10 ISBN 3-540-08663-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08663-3 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 § 64 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. 2141/3140-643210
INTRODUCTION
These notes are based on a series of lectures given at the Mathematics Institute, University of Aarhus, during the academic year 1976-77. The purpose of the lectures was to give an introduction to the classical Chern-Wei 1 theory of characteristic classes with real coefficients presupposihg only basic knowledge of differentiable manifolds and Lie groups together with elementary homology theory. Chern-Wei 1 theory is the proper generalization to higher dimensions of the classical Gauss-Bonnet theorem which states that for
M
a compact surface of genus
( 1)
where
g
in 3-space
2 (1-g)
K
is the Gaussian curvature. In particular
topological invariant of
M.
JM
K
In higher dimensions where
a compact Riemannian manifold,
K
in (1)
is a M
is
is replaced by a
closed differential form (e.g. the Pfaffian or one of the Pontrjagin forms, see chapter
examples 1 and 3)
associated to
the curvature tensor and the integration is done over a singular chain in
M.
In this way there is defined a singular cohomology
class (e.g. the Euler class or one of the Pontrjagin classes) which turns out to be a differential topological invariant in the sense that it depends only on the tangent bundle of
M
considered as a topological vector bundle. Thus a repeating theme of this theory is to show that certain quantities which
a
priori depend on the local differential
geometry are actually global topological invariants. Fundamental
IV
in this context is of course the de Rham theorem which says that every real cohomology class of a manifold
M
can be re-
presented by integrating a closed form over singular chains and on the other hand if integration of a closed form over singular chains represents the zerococycle then the form is exact. In chapter 1 we give an elementary proof of this theorem (essentially due to A. Weil [34]) which depends on 3 basic tools used several times through the lectures: gration operator of the Poincare lemma, covering,
(i) the inte-
(ii) the nerve of a
(iii) the comparison theorem for d
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