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 (hence the name "exotic").
The first exotic spheres were constructed by in dimension formula_1 as formula_2-bundles over formula_3. 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, formula_4, is the set of all ("n"+1)-tuples formula_5 of real numbers, such that the sum formula_6. For instance, formula_7 is a circle, while formula_8 is the surface of an ordinary ball of radius one in 3 dimensions. 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 formula_4 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 formula_10.
In differential topology, a more stringent condition is added, that the functions matching points in "X" with points in formula_4 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 formula_12, this monoid is a group and is isomorphic to the group formula_13 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.
Parallelizable manifolds.
The group formula_13 has a cyclic subgroup
represented by "n"-spheres that bound parallelizable manifolds. The structures of formula_15 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 formula_15 is a cyclic group, and is trivial or order 2 except in case formula_19, 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 formula_20 mod 4 is not of the form formula_21. It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all "n" bigger than 126; the case formula_22 is still open. The order of formula_23 for formula_24 is
where "B" is the numerator of formula_26, and formula_27 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 formula_17 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 formula_30 is the "n"th stable homotopy group of spheres, and "J" is the image of the "J"-homomorphism. As with formula_15, the image of "J" is a cyclic group, and is trivial or order 2 except in case formula_19, in which case it can be large, with its order related to the Bernoulli numbers. The quotient group formula_33 is the "hard" part of the stable homotopy groups of spheres, and accordingly formula_17 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.
, the Kervaire invariant problem is almost completely solved, with only the case formula_35 remaining open; see that article for details. This is primarily the work of , which proved that such manifolds only existed in dimension formula_36, and , which proved that there were no such manifolds for dimension formula_37 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 formula_13.
The order of the group formula_13 is given in this table from (except that the entry for formula_40 is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works).
Note that for dim formula_41, then formula_42 are formula_43, formula_44, formula_45, and formula_46. 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.
Milnor's construction.
One of the first examples of an exotic sphere found by was the following. Let formula_47 be the unit ball in formula_48, and let formula_2 be its boundary—a 3-sphere which we identify with the group of unit quaternions. Now take two copies of formula_50, each with boundary formula_51, and glue them together by identifying formula_52 in the first boundary with formula_53 in the second boundary. The resulting manifold has a natural smooth structure and is homeomorphic to formula_54, but is not diffeomorphic to formula_54. 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.
Brieskorn spheres.
As shown by (see also ) the intersection of the complex manifold of points in formula_56 satisfying
with a small sphere around the origin for formula_58 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres.
Twisted spheres.
Given an (orientation-preserving) diffeomorphism formula_59, gluing the boundaries of two copies of the standard disk formula_60 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. 
Setting formula_61 to be the group of twisted "n"-spheres (under connect sum), one obtains the exact sequence
For formula_63, 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. (In contrast, in the piecewise linear setting the left-most map is onto via radial extension: every piecewise-linear-twisted sphere is standard.) The group formula_61 of twisted spheres is always isomorphic to the group formula_13. The notations are different because it was not known at first that they were the same for formula_66 or 4; for example, the case formula_66 is equivalent to the Poincaré conjecture.
In 1970 Jean Cerf proved the pseudoisotopy theorem which implies that formula_68 is the trivial group provided formula_69, and so formula_70 provided formula_69.
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 Γ"k" 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:
4-dimensional exotic spheres and 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 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 "S"4 and gluing it back in using a diffeomorphism of its boundary "S"2×"S"1. The result is always homeomorphic to "S"4. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, , , , , , , , .