Exotic spheres
Contents |
1 Introduction
I should like to start this exotic spheres page by a link to my exotic spheres home page.
This already has a large collection of original source material, but some of it would be inappropriate to put on the Manifold Atlas
directly.
Andrew Ranicki
2 Construction and examples
Sphere bundles
The first known examples of exotic spheres were discovered by Milnor in [Milnor1956]. They are the total spaces of certain 3-sphere bundles over the 4-sphere as we now explain: the group parametrises linear -sphere bundles over ...
A little later Shimada [Shimada1957] used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres.
By Adams' solution of the Hopf-invariant 1 problem, [Adams1960], dimensions n = 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for m<n.
2.1 Plumbing
2.2 Twisting
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3 Invariants
Signature, Kervaire invaiant, -invariant
4 Classification
[Kervaire&Milnor1963], [Levine1983]
5 Further discussion
... is welcome
6 References
- [Adams1960] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR0141119 (25 #4530) Zbl 0096.17404
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Levine1983] J. P. Levine, Lectures on groups of homotopy spheres, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., 1126 (1983), 62–95. MR802786 (87i:57031) Zbl 0576.57028
- [Milnor1956] J. Milnor, On manifolds homeomorphic to the -sphere, Ann. of Math. (2) 64 (1956), 399–405. MR0082103 (18,498d) Zbl 0072.18402
- [Shimada1957] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59–69. MR0096223 (20 #2715) Zbl 0145.20303
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