Exotic sphere
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In mathematics, an exotic sphere is a differentiable manifold, M, that is homeomorphic to the ordinary sphere, but not diffeomorphic. That means that M is a sphere from a topological point of view, but not from the point of view of its differential structure. Thus if M has dimension n, there is a homeomorphism
- h : M → Sn
but no such h is a diffeomorphism.
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[edit] History
The first exotic spheres were constructed by John Milnor in the case n = 7. They were S3-bundles over S4. This type of exotic sphere is called a Milnor sphere. Later techniques based on surgery theory enabled calculations of the numbers of distinct exotic spheres, in any given dimension. For dimension 7, there are 28 or 15 depending on whether the differential equivalence includes orientation or not. In any dimension the classes of exotic spheres form a group under connected sum (assuming orientation counts). In the seven-dimensional case it is cyclic of order 28.
[edit] The number of exotic spheres in a given dimension
The formula of Michel Kervaire and John Milnor for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds for 
involves Bernoulli numbers. If B is the numerator of B4n/n, then
- 22n − 2(1 − 22n − 1)B
is the number of such exotic spheres. (The formula in the topological literature differs because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)
The quotient group has a description in terms of homotopy theory (the J-homomorphism).
The order of the group of smooth structures on an oriented sphere in n dimensions is given in this table.
| Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Structures | 1 | 1 | 1 | ? | 1 | 1 | 28 | 2 | 8 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 |
In dimension 4 almost nothing is known about the group of smooth structures, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite. In dimensions other than 4 the group is finite (and abelian).
[edit] Explicit examples
Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7.
The intersection of the complex manifold of points in C5 satisfying
- a2 + b2 + c2 + d3 + e6k − 1 = 0
with a small sphere around the origin for k = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. See the book by Hirzebruch and Mayer for details.
[edit] Gluck twists
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. Some candidates for such structures are given by Gluck twists. These are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. But in most cases it is unknown whether or not the result is diffeomorphic to S4. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S4, but there are plently of other ways to knot a 2-sphere in S4.)
- Herman Gluck, The embedding of two-spheres in the four-sphere,, Trans. Amer. Math. Soc. 104 (1962), 308-333.
[edit] See also
[edit] References
- On Manifolds Homeomorphic to the 7-Sphere, John Milnor, The Annals of Mathematics, 2nd Ser., Vol. 64, No. 2. (Sep., 1956), pp. 399-405. This gives the first examples of exotic spheres.
- Differentiable Structures on Spheres, John Milnor, American Journal of Mathematics, Vol. 81, No. 4. (Oct., 1959), pp. 962-972.
- Groups of Homotopy Spheres: I Michel A. Kervaire; John W. Milnor, The Annals of Mathematics, 2nd Ser., Vol. 77, No. 3. (May, 1963), pp. 504-537. This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4.
- Hirzebruch and Mayer, O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten, Springer lecture notes in mathematics 57. This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
- Number of h-cobordism classes of smooth homotopy n-spheres. For n not equal to 4 this is the number of oriented diffeomorphism classes of differentiable structures on the n-sphere.
