Exotic sphere

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In mathematics, an exotic sphere is a differentiable manifold that is homeomorphic to the standard Euclidean n-sphere, but not diffeomorphic. That means that such a manifold 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.

The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4. He showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4.

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[edit] The monoid of smooth structures on spheres in a given dimension

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 operation. 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 Michael Freedman in dimension 4, Stephen Smale in higher dimensions, and Grigori Perelman in dimension 3. In dimension 3, Edwin Moise proved that every topological manifold has an essentially unique smooth structure, so the monoid of smooth structures on the 3-sphere is trivial.

The group Θn has a cyclic subgroup

bPn+1

represented by n-spheres that bound parallelizable manifolds. The structures of bPn+1 and the quotient

Θn/bPn+1

are described separately in the paper (Michel Kervaire & John Milnor 1963).

The group bPn+1 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

2^{2n-2}(2^{2n-1}-1)B \,\!

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.)

The quotient group Θn/bPn+1 has a description in terms of stable homotopy groups of spheres modulo the image of the J-homomorphism). More precisely there is an injective map

Θn/bPn+1 → πnS/J

where πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. Browder (1969) proved this is an isomorphism if n is not of the form 2k−2, and if n is of this form its image is either the whole group or a subgroup of index 2, and is a subgroup of index 2 in the first few cases when n is 2, 6, 14, 30, or 62.

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).

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

Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres.

[edit] Explicit examples of exotic spheres

One of the first examples of an exotic sphere found by Milnor was the following: 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. Milnor said about these examples: "When I came upon such an example in the mid-50’s, I was very puzzled and didn’t know what to make of it. At first, I thought I’d found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to S7. Thus, there exists a differentiable structure on S 7 not diffeomorphic to the standard one."

As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in C5 satisfying

a2 + b2 + c2 + d 3 + 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.

[edit] Twisted spheres

Given an (orientation-preserving) diffeomorphism f: Sn−1→Sn−1, gluing the boundaries of two copies of the standard disk Dn together by f yields a manifold called a twisted sphere (with twist f). It is homotopy equivalent to the standard n-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. (Milnor 1959b) Setting Γn to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence

\pi_0\,\text{Diff}^+(D^n) \to \pi_0\,\text{Diff}^+(S^{n-1}) \to \Gamma_n \to 0 \,\!

For n > 4, every exotic sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale. (In contrast, in the PL setting, via radial extension the left-most map is onto: there are no PL-twisted spheres.) The group Γn of twisted spheres is always isomorphic to the group Θn. The notations are different, because it was not known at first that they were the same for n=3 or 4; for example, the case n=3 is equivalent to the Poincare conjecture.

[edit] Applications

If M is a PL-manifold, then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γk = Θk. More precisely, the obstructions to the existence of any smooth structure lie in the groups Hk+1(M, Γk) for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups Hk(M, Γk). In particular the groups Γk vanish if k<7, so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6.

The following finite abelian groups are essentially the same:

  • The group Θn of h-cobordism classes of oriented homotopy n-spheres.
  • The group of h-cobordism classes of oriented n-spheres.
  • The group Γn of twisted oriented n spheres.
  • The homotopy group πn(PL/DIFF)
  • If n≠3, the homotopy πn(TOP/DIFF) (if n=3 this group has order 2; see Kirby-Siebenmann invariant).
  • The group of smooth structures of an oriented PL n-sphere.
  • If n≠4, the group of smooth structures of an oriented topological n-sphere.

[edit] Gluck twists

In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincare conjecture". Some candidates for such structures are given by Gluck twists (Gluck 1962). 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.)

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