Topics in the Homology Theory of Fibre Bundles: Lectures Given at the University of Chicago, 1954 (Lecture Notes in Mathematics, 36) [1967 ed.] 3540039074, 9783540039075
A Collection of informal reports and seminars
126 94 612KB
English Pages 98 [99] Year 1967
Table of contents :
Title page
INTRODUCTION
TABLE OP CONTENTS
Chapter I. The homologlcal properties of H-spaces.
1. Algebraic preliminaries.
2. Topological preliminaries.
3. Structure theorem for Hopf algebras.
4. Primitive elements, Samelson's theorem.
5. The Pontrjagin product.
6. Homology of H-spaces.
7. Spaces on which an H-space operates.
Bibliography
Chapter II. Spectral sequence of a fibre bundle.
8. Differential and filtered modules.
9. Notion of a spectral sequence.
10. Spectral sequence of a differential filtered module.
11. Systems of local coefficients.
12. Fibre bundles.
13. Spectral sequences of a fibre bundle.
14. Some simple applications.
15. Pairing of the spectral sequence of a principal bundle with the homology of the structural group.
Bibliography.
Chapter III. Universal bundles and classifying spaces.
16. Universal bundles and classifying spaces.
17. p(U,G) : three fiberings involving classifying spaces.
18. Some results on universal spectral sequences.
19. Proof of one theorem on universal spectral sequences.
20. Invariants of the Weyl group and classifying spaces. The Hirsch formula.
Bibliography.
Chapter IV. Classifying spaces of the classical groups.
21. Unitary groups.
22. Orthogonal groups, cohomology mod 2.
23. Orthogonal groups, cohomology mod p i 2.
24. Integral cohomology of Bq/n and Bsn(n)
25. Stiefel-Whitney classes, Pontrjagin classes.
Bibliography.
Bibliographical notes and comments.
Title page
INTRODUCTION
TABLE OP CONTENTS
Chapter I. The homologlcal properties of H-spaces.
1. Algebraic preliminaries.
2. Topological preliminaries.
3. Structure theorem for Hopf algebras.
4. Primitive elements, Samelson's theorem.
5. The Pontrjagin product.
6. Homology of H-spaces.
7. Spaces on which an H-space operates.
Bibliography
Chapter II. Spectral sequence of a fibre bundle.
8. Differential and filtered modules.
9. Notion of a spectral sequence.
10. Spectral sequence of a differential filtered module.
11. Systems of local coefficients.
12. Fibre bundles.
13. Spectral sequences of a fibre bundle.
14. Some simple applications.
15. Pairing of the spectral sequence of a principal bundle with the homology of the structural group.
Bibliography.
Chapter III. Universal bundles and classifying spaces.
16. Universal bundles and classifying spaces.
17. p(U,G) : three fiberings involving classifying spaces.
18. Some results on universal spectral sequences.
19. Proof of one theorem on universal spectral sequences.
20. Invariants of the Weyl group and classifying spaces. The Hirsch formula.
Bibliography.
Chapter IV. Classifying spaces of the classical groups.
21. Unitary groups.
22. Orthogonal groups, cohomology mod 2.
23. Orthogonal groups, cohomology mod p i 2.
24. Integral cohomology of Bq/n and Bsn(n)
25. Stiefel-Whitney classes, Pontrjagin classes.
Bibliography.
Bibliographical notes and comments.
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