The Topology of Stiefel Manifolds

I. M. James

doi:10.1017/CBO9780511600753

Stiefel manifolds are an interesting family of spaces much studied by algebraic topologists. These notes, which originated in a course given at Harvard University, describe the state of knowledge of the subject, as well as the outstanding problems. The emphasis throughout is on applications (within the subject) rather than on theory. However, such theory as is required is summarized and references to the literature are given, thus making the book accessible to non-specialists and particularly graduate students. Many examples are given and further problems suggested.

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The Topology of Stiefel Manifolds

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

1 - Introduction: algebra versus topology
pp 1-12

2 - The Stiefel manifolds
pp 13-20

3 - The auxiliary spaces
pp 21-26

4 - Retractible fibrations
pp 27-32

5 - Thom spaces
pp 33-39

6 - Homotopy equivariance
pp 40-44

7 - Cross-sections and the S-type
pp 45-51

8 - Relative Stiefel manifolds
pp 52-56

9 - Cannibalistic characteristic classes
pp 57-61

10 - Exponential characteristic classes
pp 62-70

11 - The main theorem of J-theory
pp 71-77

12 - The fibre suspension
pp 78-82

13 - Canonical automorphisms
pp 83-88

14 - The iterated suspension
pp 89-93

15 - Samelson products
pp 94-99

16 - The Hopf construction
pp 100-108

17 - The Bott suspension
pp 109-115

18 - The intrinsic join again
pp 116-122

19 - Homotopy-commutativity
pp 123-128

20 - The triviality problem
pp 129-132

21 - When is Pn, k neutral?
pp 133-138

22 - When is Vn, 2 neutral?
pp 139-142

23 - When is Vn, k neutral?
pp 143-149

24 - Further results and problems
pp 150-154

Bibliography
pp 155-166

Index
pp 167-168

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