Сфералардың ерекше құрылымдары және дифференциалдық топология

Smooth manifold that is homeomorphic but not diffeomorphic to a sphere

In an area of mathematics called 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 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. More generally, in any dimension n ≠ 4, there is a finite Abelian group whose elements are the equivalence classes of smooth structures on Sn, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y] = [x + y], where x and y are arbitrary representatives of their equivalence classes, and x + y denotes the smooth structure on the smooth Sn that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.

The unit n sphere, , is the set of all (n+1) tuples of real numbers, such that the sum For instance, is a circle, while is the surface of an ordinary ball of radius one in 3 dimensions. Topologists consider a space X to be an n sphere if there is a homeomorphism between them, i. e. every point in X may be assigned to exactly one point in the unit n sphere by a continuous bijection with continuous inverse. For example, a point x on an n sphere of radius r can be matched homeomorphically with a point on the unit n sphere by multiplying its distance from the origin by Similarly, an n cube of any radius is homeomorphic to an n sphere. In differential topology, two smooth manifolds are considered smoothly equivalent if there exists a diffeomorphism from one to the other, which is a homeomorphism between them, with the additional condition that it be smooth — that is, it should have derivatives of all orders at all its points — and its inverse homeomorphism must also be smooth. 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. No exotic structures are possible on the 1 , 2 , 3 , 5 , 6 , 12 , 56 or 61 sphere. 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.

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 (see Moise's theorem), so the monoid of smooth structures on the 3 sphere is trivial.

The quotient group 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 is the nth stable homotopy group of spheres, and J is the image of the J homomorphism. As with , 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 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.

One of the first examples of an exotic sphere found by was the following. Let be the unit ball in , and let be its boundary—a 3 sphere which we identify with the group of unit quaternions. Now take two copies of , each with boundary , and glue them together by identifying in the first boundary with in the second boundary. The resulting manifold has a natural smooth structure and is homeomorphic to , but is not diffeomorphic to 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.

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, , , , , , , , .