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.
2 Construction and examples
Exotic spheres may be constructed in a variety of ways.
2.1 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. In [Brieskorn] and [Brieskorn] it is shown in particular that all homotopy spheres in and can be realised as follows. Let be a string of 2's in a row with , then there are diffeomorphisms
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 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.
2.3 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 , 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 are isomorphic to the cokernel of the J-homomorphism, .
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
- [Brieskorn] Template:Brieskorn
- [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
- [Eells&Kuiper1962] J. Eells and N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. MR0156356 (27 #6280) Zbl 0119.18704
- [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
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- [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
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- [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
- [Wall1962a] C. T. C. Wall, Classification of -connected -manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022