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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 , following [Milnor1968] we define the closed smooth oriented -connected -manifold
The manifolds are often called Brieskorn varieties. By construction, every lies in and so bounds a parallelisable manifold. In [Brieskorn1966, Korollar 2] (see also [Brieskorn1966a] and [Hirzebruch&Mayer1968]), it is shown 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
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, p.583, The group of diffeomorphisms of ].
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 diffeomorphism of . In this way we obtain a bilinear pairing
In particular for we see that generates .
 3 Invariants
Finding invariants of exotic sphere which distinguish it from the standard sphere is rather a subtle undertaking. Moreover such invariants are often defined via a manifold with . In this case finding an intrinsic definition and or computation of the relevant invariant can also be subtle.
We begin by listing some invariants which are equal for all exotic spheres.
Proposition 3.1. Let be a closed smooth manifold homeomorphic to the n-sphere. Then
- there is an isomorphism of tangent bundles ,
- the signature of vanishes,
- the Kervaire invariant of is zero for every framing of .
(To make sense of the first statement remember that the topological space underlying every exotic sphere is homeomorphic to .)
Remark 3.2. The analogue of the first statement for the stable tangent bundle was proven in [Kervaire&Milnor1963, Theorem 3.1]. A proof of the unstable statement is given in [Ray&Pedersen1980, Lemma 1.1]. The next two statements are obvious since both the signature and Kervaire invariant are defined to be zero if and via a symmetric or quadratic form on if .
 3.1 Bordism classes
As every homotopy sphere is stably parallelisable, homotopy spheres admit -structures for any . If is such that for any stable framing of , then we obtain a well-defined homomorphism
- If for then is isomorphic to almost framed bordism and the homomorphism is the same thing as the in Theorem 4.1.
- Perhaps surprisingly for all , as explained in the next subsection.
- In general determining is a hard and interesting problem.
- -coboundaries for elements of are often used to define invariants of -null bordant homotopy spheres.
 3.2 The α-invariant
In dimensions , every exotic sphere has a unique Spin structure and from above we have the homomorphism . Recall the -invariant homomorphism and that there are isomorphisms for all .
Theorem 3.3 [Anderson&Brown&Peterson1967]. We have if and only if and if and only if or .
 3.3 The Eells-Kuiper invariant
 3.4 The s-invariant
 4 Classification
For and , . For , is unknown. We therefore concentrate on higher dimensions.
Here is the n-th L-group of the the trivial group: as n = 0, 1, 2 or 3 modulo 4 and the sequence ends at . Also is the stable orthogonal group and is the stable group of homtopy self-equivalences of the sphere. There is a fibration and the groups fit into the homtopy long exact sequence
of this fibration. The homomorphism is the stable J-homomorphism. In particular, by [Serre1951] the groups are finite and by [Bott1959], [Adams1966] and [Quillen1971] the domain, image and kernel of have been completely determined. An important result in [Kervaire&Milnor1963] is that the homomorphism is nonzero. The above sequence then gives
Theorem 4.1 [Kervaire&Milnor1963]. For , the group is finite. Moreover there is an exact sequence
where , the group of homotopy spheres bounding paralellisable manifolds, is a finite cyclic group which vanishes if is even. Moreover unless when it is or .
The groups are known for up to approximately 62. In general their determination is a very hard problem. Modulo this problem we see two remaining problems in the determination of : an extension problem and the comptutation of the order of the groups and . We discuss these in turn.
The map is the Kervaire invariant and by definition . By the long exact sequence above we have
Theorem 4.3 [Kervaire&Milnor1963, Section 8]. The group is either or . Moreover the following are equivalent:
- the Kervaire sphere is diffeomorphic to the standard sphere,
- there is a framed manifold with Kervaire invariant 1: .
Conversely the following are equivalent:
- the Kervaire sphere is not diffeomorphic to the standard sphere,
- there is no framed manifold with Kervaire invariant 1: .
 4.1 The order of bP4k
The group is a cyclic group whose order can be determined using the Hirzebruch's signature theorem if one knows the order of . Adams determined the latter group up to a factor of two which was settled by Quillen with a positive solution to the Adams conjecture.
Theorem 4.4. Let , let be the k-th Bernoulli number (topologist indexing) and for let denote the numerator of expressed in lowest form. Then for , the order of is
Remark 4.5. Note that is odd so the 2-primary order of is while the odd part is . Modulo the Adams conjecture the proof appeared in [Kervaire&Milnor1963, Section 7]. Detailed treatments can also be found in [Levine1983, Section 3] and [Lück2001, Chapter 6].
The following table lists factorisations of for .
4k 8 12 16 20 24 28 32 order bP4k 22.7 25.31 26.127 29.511 210.2047.691 213.8191 214.16384.3617
 4.2 The order of bP4k+2
The situation for which is now almost completely understood as well. References for the theorem are given in the remark which follows it.
Theorem 4.6. The group is given as follows:
- or ,
Remark 4.7. The following is a chronological list of determinations of :
 5 Further discussion
 5.1 Curvature on exotic spheres
Gromoll-Meyer proved that a certain exotic 7-sphere can be realized as a biquotient of the compact Lie group Sp(2) and thus by the O'Neill formula has a Riemannian metric of nonnegative sectional curvature. It is not known whether there exist exotic spheres with Riemannian metrics of positive sectional curvature. For a recent review of which exotic spheres admit metrics of various sorts of positive curvature see [Joachim&Wraith2008].
 5.2 The Kervaire-Milnor braid
 6 Topological manifolds admitting no smooth structure
If is exotic then it turns out that is a topological manifold which admits no smooth structure.
[Kervaire1960a] shows that is non-smoothable and the arugments there work for all odd so long as the Kervaire sphere is exotic.
When is even the proof is more complicated: one first need's Novikov's theorem that the rational Pontrjagin classes of a topological manifold are homeomorphism invariants [Novikov1965b]. Prior to Novikov's theorem, some weaker statements were known. For example, when and is the total space of a -bundle over as above and if then by [Tamura1961] is smoothable if and only if mod . ; Applying Novikov's theorem we now know that is smoothable if and only if mod .
 7 Footnotes
- ↑ Note that Tamura uses a different identification from the one used above.
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- The Wikipedia page on exotic spheres
- The tabulation of the order of the group of exotic spheres in the On-Line Encyclopedia of Integer Sequences
- Bulletin of the AMS Volume 52 Number 4 Volume focusing on smooth structures on manifolds, in particular the work of Kervaire and Milnor
- Andrew Ranicki's exotic sphere home page, with many of the original papers: http://www.maths.ed.ac.uk/~aar/exotic.htm
- Including some original correspondence between Kervaire and Milnor
- An animation of exotic 7-spheres. Slides from a presentation by Nile Johsnon at the Second Abel conference in honor of John Milnor.