Exotic spheres

Revision as of 14:32, 21 November 2009

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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; I should like to start this exotic spheres page by a [http://www.maths.ed.ac.uk/~aar/exotic.htm 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. <br> Andrew Ranicki </wikitex> == Construction and examples == <wikitex>; {{cite|Milnor1956}} </wikitex> == Invariants == <wikitex>; Signature, Kervaire invaiant, $\alpha$-invariant </wikitex> == Classification == <wikitex>; {{cite|Kervaire&Milnor1963}}, {{cite|Levine1983}} </wikitex> == Further discussion == <wikitex>; ... is welcome </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\pi_3(SO(4)) \cong \Zz \oplus \Zz$ parametrises linear $3$-sphere bundles over $S^4$ ...

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

</wikitex>

3 Invariants

Signature, Kervaire invaiant, $\alpha$-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 $7$-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