Exotic sphere


In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.
The first exotic spheres were constructed by in dimension as -bundles over. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension 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. The classification of exotic spheres by 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.

Introduction

The unit n-sphere,, is the set of all -tuples of real numbers, such that the sum. Topologists consider a space, X, to be an n-sphere if every point in X can be assigned to exactly one point in the unit n-sphere in a continuous way, which means that sufficiently nearby points in X get assigned to nearby points in Sn and vice versa. For example, a point x on an n-sphere of radius r can be matched with a point on the unit n-sphere by adjusting its distance from the origin by.
In differential topology, a more stringent condition is added, that the functions matching points in X with points in should be smooth, that is they should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an unsolved problem.

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, this monoid is a group and is isomorphic to the group 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, so the monoid of smooth structures on the 3-sphere is trivial.

Parallelizable manifolds

The group has a cyclic subgroup
represented by n-spheres that bound parallelizable manifolds. The structures of and the quotient
are described separately in the paper, 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 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 proved that it has order 2 if mod 4 is not of the form. It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 125; the case is still open.
The order of for is
where B is the numerator of, and is a Bernoulli number.

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, 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.
, the Kervaire invariant problem is almost completely solved, with only the case remaining open; see that article for details. This is primarily the work of, which proved that such manifolds only existed in dimension, and, 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 Θ''n''

The order of the group Θn is given in this table from .
Note that for dim n = 4k-1, then Θn are 28 = 22, 992 = 25, 16256 = 27, and 523264 = 210. Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres.
By computations of stable homotopy groups of spheres, proves that the sphere has a unique smooth structure, and it is the last odd dimensional one -- the only ones are,,, and.

Explicit examples of exotic spheres

One of the first examples of an exotic sphere found by was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying in the boundary with,. The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.
As shown by the intersection of the complex manifold of points in C5 satisfying
with a small sphere around the origin for k = 1, 2,..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres.

Twisted spheres

Given an diffeomorphism, gluing the boundaries of two copies of the standard disk Dn together by f yields a manifold called a twisted sphere. It is homotopy equivalent to the standard n-sphere because the gluing map is homotopic to the identity, but not in general diffeomorphic to the standard sphere.
Setting to be the group of twisted n-spheres, one obtains the exact sequence
For, every exotic n-sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale which can be seen as a consequence of the h-cobordism theorem. 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 ; for example, the case is equivalent to the Poincaré conjecture.
In 1970 Jean Cerf proved the pseudoisotopy theorem which implies that is the trivial group provided, so provided.

Applications

If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups. More precisely, the obstructions to the existence of any smooth structure lie in the groups for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups.
In particular the groups Γk vanish if, 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:
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 Poincaré conjecture", and is discussed by who say that it is believed to be false.
Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres and those derived by Gluck twists. Gluck twist spheres 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. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example,,,,,,,,.