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").
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.
The classification of exotic spheres by Michel Kervaire and Milnor (1963) 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.
To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians (including Milnor himself) were surprised in 1956 when Milnor showed that consistent local 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.
of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian.
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.
has a cyclic subgroup represented by n-spheres that bound parallelizable manifolds.
and the quotient are described separately in the paper (Kervaire & 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.
It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 126; the case
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.
is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism.
Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.
remaining open, although Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University, on May 30, 2024, that the final case of dimension 126 has been settled and that there exist manifolds of Kervaire invariant 1 in dimension 126.
Previous work of Browder (1969), proved that such manifolds only existed in dimension
is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works).
At first, I thought I'd found a counterexample to the generalized Poincaré conjecture in dimension seven.
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 Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in
to be the group of twisted n-spheres (under connect sum), 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.
(In contrast, in the piecewise linear setting the left-most map is onto via radial extension: every piecewise-linear-twisted sphere is standard.)
If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γk = Θk.
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 Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
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.
Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture.
For example, Cameron Gordon (1976), José Montesinos (1983), Steven P. Plotnick (1984), Gompf (1991), Habiro, Marumoto & Yamada (2000), Selman Akbulut (2010), Gompf (2010), Kim & Yamada (2017).