Topics in the Homology Theory of Fibre Bundles Lectures given at the
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36 Armand Borel The Institute for Advanced Study Princeton, New Jersey, USA
Topics in the Homology Theory of Fibre Bundles Lectures given at the University of Chicago, 1954 Notes by Edward Halpern
1967
Springer-Verlag· Berlin Heidelberg· New York
All rights, especially that of translation into foreign languages, reserved, It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1967. Library of Congress Catalog Card Number 67-28561 Printed in Germany. Tide No. 7356.
INTRODUCTION This fascicle consists of the Notes of a course given at the University of Chicago in 1954. the purpose of which was to discuss some then recent developments in the homology theory of fibre bundles. pertaining to H-spaces. spectral sequences. classifying spaces and characteristic classes. Since then. of course. alternative approaches to some of these topics have been introduced. and new results have been obtained. which make: these Notes outdated in several respects. In spite (or maybe because) of this. it was recently suggested that they be included in this series. No changes have been made in the original version. written by E. Halpern. to whom I am glad to express my hearty thanks. However. some comments have been added at the end. which. without aiming at completeness. point out further results and give more recent references. A. Borel
TABLE OF CONTENTS Chapter 1. 1. 2.
3. 4.
5.
6. 7.
The homological properties of H-spaces. Algebraic preliminaries. Topological preliminaries. Structure theorem for Hopf algebras. Primitive elements, Samelson's theorem. The Pontrjagin product. Homology of H-spaces. Spaces on which an H-space operates. Bibliography
Chapter II.
8. 9.
10. 11. 12.
13. 14. 15.
1. 2.
4.
11.
14. 17. 19.
25.
Spectral sequence of a fibre bundle.
Differential and filtered modules. Notion of a spectral sequence. Spectral sequence of a differential filtered module. Systems of local coefficients. Fibre bundles. Spectral sequences of a fibre bundle. Some simple applications. Pairing of the spectral sequence of a principal bundle with the homology of the structural group. Bibliography.
26.
28. 29.
31.
33. 37. 41.
46.
51.
Chapter III. Universal bundles and classifying spaces.
16. 17. 18. 19. 20.
Universal bundles and classifying spaces. p(U,G) : three fiberings involving classifying spaces. Some results on universal spectral sequences. Proof of one theorem on universal spectral sequences. Invariants of the Weyl group and classifying spaces. The Hirsch formula. Bibliography.
Chapter IV. 21. 22. 23.
24. 25.
52. 55. 56.
57.
64. 70.
Classifying spaces of the classical groups.
Unitary groups. Orthogonal groups, cohomology mod 2. Orthogonal groups, cohomology mod p 2. Integral cohomology of BO(n) and BSO(n). Stiefel-Whitney classes, Pontrjagin classes. Bibliography.
Bibliographical notes and comments.
71.
75. 82.
84
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