Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction: algebra versus topology
- 2 The Stiefel manifolds
- 3 The auxiliary spaces
- 4 Retractible fibrations
- 5 Thom spaces
- 6 Homotopy equivariance
- 7 Cross-sections and the S-type
- 8 Relative Stiefel manifolds
- 9 Cannibalistic characteristic classes
- 10 Exponential characteristic classes
- 11 The main theorem of J-theory
- 12 The fibre suspension
- 13 Canonical automorphisms
- 14 The iterated suspension
- 15 Samelson products
- 16 The Hopf construction
- 17 The Bott suspension
- 18 The intrinsic join again
- 19 Homotopy-commutativity
- 20 The triviality problem
- 21 When is Pn, k neutral?
- 22 When is Vn, 2 neutral?
- 23 When is Vn, k neutral?
- 24 Further results and problems
- Bibliography
- Index
21 - When is Pn, k neutral?
Published online by Cambridge University Press: 23 December 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction: algebra versus topology
- 2 The Stiefel manifolds
- 3 The auxiliary spaces
- 4 Retractible fibrations
- 5 Thom spaces
- 6 Homotopy equivariance
- 7 Cross-sections and the S-type
- 8 Relative Stiefel manifolds
- 9 Cannibalistic characteristic classes
- 10 Exponential characteristic classes
- 11 The main theorem of J-theory
- 12 The fibre suspension
- 13 Canonical automorphisms
- 14 The iterated suspension
- 15 Samelson products
- 16 The Hopf construction
- 17 The Bott suspension
- 18 The intrinsic join again
- 19 Homotopy-commutativity
- 20 The triviality problem
- 21 When is Pn, k neutral?
- 22 When is Vn, 2 neutral?
- 23 When is Vn, k neutral?
- 24 Further results and problems
- Bibliography
- Index
Summary
This section is based on joint work with Sutherland. Recall that dn denotes the self-map of Pn,k defined by reflection in the last coordinate hyperplane. We say that Pn, k is neutral (elsewhere outsimple) if dn ≃ 1, and define S-neutral similarly. If n and k are both odd then Pn,k is neutral, as remarked in §7. If n is even then dn has degree −1 on the integral homology Hn−1,(Pn,k) = Z, and so Pn,k is not S-neutral. Thus the interest resides in the case when n is odd and k even. Notice that Pk+1,k = Pk is neutral for all even values of k. In the course of §6 we have already proved
Proposition (21.1). Suppose that Pn,k is S-neutral, where n is odd and k even. Then Pm+n,kand Pm+k−n,kare S-neutral, whenever m ≡ 0 mod âk.
Here âk is as in (1.10). Now consider Vn,k as a Z2 -space under the outer automorphism which changes the sign of the last row and column of each matrix. Since the inclusion Pn,k → Vn,k is a Z2 - map it follows at once from (3. 4) that Pn,k is neutral if Vn,k is neutral and n ≥ 2k.
- Type
- Chapter
- Information
- The Topology of Stiefel Manifolds , pp. 133 - 138Publisher: Cambridge University PressPrint publication year: 1977