Exotic spheres
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1 Introduction
By a homotopy sphere we mean a closed smooth oriented 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. An important subgroup of is which consists of those homotopy spheres which bound parallelisable manifolds.
2 Construction and examples
The first exotic spheres discovered were certain 3-sphere bundles over the 4-sphere, [Milnor1956]. Following this discovery there was a rapid development of techniques which construct exotic spheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting.
2.1 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 .
2.2 Brieskorn varieties
Let be a point in and let be a string of n+1 positive integers. Given the complex variety and the -sphere for small , we define the closed smooth oriented (2n-1)-manifolds
The manifolds are often called Brieskorn varieties. By construction, every lies in and so bounds a parallelisable manifold. In [Brieskorn1966] and [Brieskorn1966a] (see also [Hirzebruch&Mayer1968]), it is shown in particular that all homotopy spheres in and can be realised as for some . Let be a string of 2k-1 2's in a row with , then there are diffeomorphisms
2.3 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 where a pair gives rise to a bundle with Euler number and first Pontrjagin class : here we orient and so identify . If we set then the long exact homotopy sequence of a fibration and Poincare duality ensure that the manifold , the total space of the bundle , is a homotopy sphere. Milnor first used a -invariant, called the -invariant, to show, e.g. that is not diffeomorphic to . A little later Kervaire and Milnor [Kervaire&Milnor1963] proved that and Eells and Kuiper [Eells&Kuiper1962] defined a refinement of the -invariant, now called the Eells-Kuiper -invariant, which in particular gives
Shimada [Shimada1957] used similar techniques to show that the total spaces of certain 7-sphere bundles over the 8-sphere are exotic 15-spheres. In this case and the bundle has Euler number and second Pontrjagin class . Moreover where the -summand is as explained below. Results of [Wall1962a] and [Eells&Kuiper1962] combine to show that
- By Adams' solution of the Hopf-invariant 1 problem, [Adams1958] and [Adams1960], the dimensions n = 3, 7 and 15 are the only dimensions in which a topological n-sphere can be fibre over an m-sphere for 0 < m < n.
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 for we see that generates .
3 Invariants
Signature, Kervaire invariant, -invariant, Eels-Kuiper invariant, -invariant.
4 Classification
For , the group of exotic n-spheres fits into the following long exact sequence, first discovered in [Kervaire&Milnor1963] (more details can also be found in [Levine1983]):
Here is the i-th L-group of the the trivial group: as i = 0, 1, 2 or 3 modulo 4 and the sequence ends at . The groups fit into a long exact sequence
where is the J-homomorphism. By [Serre1951] the groups are finite.
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
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