Classification
The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation-preserving diffeomorphism. The monoid operation is the connected sum. Provided n ≠ 4, this monoid is a group and is isomorphic to the group Θn of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on Gluck twists. All homotopy n-spheres are homeomorphic to the n-sphere by the generalized Poincaré conjecture, proved by Stephen Smale in dimensions bigger than 4, Michael Freedman in dimension 4, and Grigori Perelman in dimension 3. In dimension 3, Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure (see Moise's theorem), so the monoid of smooth structures on the 3-sphere is trivial.
- bPn+1
The group Θn has a cyclic subgroup
represented by n-spheres that bound parallelizable manifolds. The structures of bPn+1 and the quotient
are described separately in the paper (Michel Kervaire & John Milnor 1963), which was influential in the development of surgery theory. In fact, these calculations can be formulated in a modern language in terms of the surgery exact sequence as indicated here.
The group bPn+1 is a cyclic group, and is trivial or order 2 except in case in which case it can be large, with its order related to the Bernoulli numbers. It is trivial if n is even. If n is 1 mod 4 it has order 1 or 2; in particular it has order 1 if n is 1, 5, 13, 29, or 61, and Browder (1969) proved that it has order 2 if n = 1 mod 4 is not of the form 2k – 3. The order of bP4n for n ≥ 2 is
where B is the numerator of |4B2n/n|, and B2n is a Bernoulli number. (The formula in the topological literature differs slightly because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)
- Map between quotients
The quotient group Θn/bPn+1 has a description in terms of stable homotopy groups of spheres modulo the image of the J-homomorphism); it is either equal to the quotient or index 2. More precisely there is an injective map
where πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. As with bPn+1, the image of J is a cyclic group, and is trivial or order 2 except in case in which case it can be large, with its order related to the Bernoulli numbers. The quotient group is the "hard" part of the stable homotopy groups of spheres, and accordingly is the hard part of the exotic spheres, but almost completely reduces to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with index 2. The latter is the case if and only if there exists an n-dimensional framed manifold with Kervaire invariant 1, which is known as the Kervaire invariant problem. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.
As of 2012, the Kervaire invariant problem is almost completely solved, with only the case n = 126 remaining open; see that article for details. This is primarily the work of Browder (1969), which proved that such manifolds only existed in dimension n = 2j − 2, and Hill, Hopkins & Ravenel (2009), which proved that there were no such manifolds for dimension and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30, and 62, but dimension 126 is open, with no manifold being either constructed or disproven.
- Order of Θn
The order of the group Θn is given in this table (sequence A001676 in OEIS) from (Kervaire & Milnor 1963) (except that the entry for n = 19 is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works).
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Dim n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 order Θn 1 1 1 1 1 1 28 2 8 6 992 1 3 2 16256 2 16 16 523264 24 bPn+1 1 1 1 1 1 1 28 1 2 1 992 1 1 1 8128 1 2 1 261632 1 Θn/bPn+1 1 1 1 1 1 1 1 2 2×2 6 1 1 3 2 2 2 2×2×2 8×2 2 24 πnS/J 1 2 1 1 1 2 1 2 2×2 6 1 1 3 2×2 2 2 2×2×2 8×2 2 24 index - 2 - - - 2 - - - - - - - 2 - - - - - -
Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres.
Read more about this topic: Exotic Sphere