Exotic spheres
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1 Introduction
A homotopy sphere is a closed, smooth n-manifold homotopy equivalent to . The manifold is called an exotic sphere if it is not diffeomorphic to . By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension is homeomorphic to : this statement holds in dimension 2 by the classification of surfaces and was famously proven in dimension 4 in [Freedman1982] and in dimension 3 by Perelman. We define
to be the set of oriented diffeomorphism classes of homotopy spheres. Connected sum makes into an abelian group with inverse given by reversing orientation.
2 Construction and examples
Exotic spheres may be constructed in a variety of ways.
2.1 Brieskorn varieties
2.2 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, [Adams1958] and [Adams1960], the dimensions n = 3, 7 and 15 are the only dimensions where an n-sphere can be fibre over an m-sphere for 0 < m < n.
2.3 Plumbing
As special case of the following construction goes back at least to [Milnor1959].
Let , let be pairs of positive integers such that and let be the clutching functions of -bundles over
Let be a graph with vertices such that the edge set between and , is non-empty only if . We form the manifold from the disjoint union of the by identifying and for each edge in . If is simply connected then
is often a homotopy sphere. We establish some notation for graphs, bundles and define
- let denote the graph with two vertices and one edge connecting them and define ,
- let denote the -graph,
- let denote the tangent bundle of the -sphere,
- let , , denote a generator,
- let , denote a generator:
- let be the suspension homomorphism,
- for and for ,
- let be essential.
Then we have the following exotic spheres.
- , the Milnor sphere, generates , .
- , the Kervaire sphere, generates .
- is the inverse of the Milnor sphere for .
- For general , is exotic.
- , generates .
- , generates .
2.4 Twisting
By [Cerf1970] and [Smale1962a] there is an isomorphism for where is the group of isotopy classes of orientation preserving diffeomorphisms of . The map is given by
Hence one may construct exotic (n+1)-spheres by describing diffeomorphisms of which are not isotopic to the identity. We give such a construction which probably goes back to Milnor: so far the earliest reference found is the problem list of the 1963 Seattle topology conference [Lashof1965].
Represent and by smooth compactly supported functions and and define the following self-diffeomorphisms of
If follows that is compactly supported and so extends uniquely to a diffeomrphism of . In this way we obtain a bilinear pairing
such that
In particular generates for .
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant, -invariant.
4 Classification
For , group of exotic spheres fits into the following long exact sequence:
The existence of the above sequence was proven in [Kervaire&Milnor1963]. More details can also be found in [Levine1983].
5 Further discussion
6 External references
- Wikipedia article on exotic spheres
- http://www.maths.ed.ac.uk/~aar/exotic.htm Andrew Ranicki's exotic sphere home page, with many of the original papers.
7 References
- [Adams1958] J. F. Adams, On the nonexistence of elements of Hopf invariant one, Bull. Amer. Math. Soc. 64 (1958), 279–282. MR0097059 (20 #3539) Zbl 0178.26106
- [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
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [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
- [Lashof1965] R. Lashof, Problems in differential and algebraic topology. Seattle Conference, 1963, Ann. of Math. (2) 81 (1965), 565–591. MR0182961 (32 #443) Zbl 0137.17601
- [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
- [Milnor1959] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR0110107 (22 #990) Zbl 0111.35501
- [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
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103