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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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Topological equivalence redirects here see also Topological equivalence (dynamical systems. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable 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. In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential 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. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential 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. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, specifically in Topology, the operation of connected sum is a geometric modification on Manifolds Its effect is to join two given manifolds 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. In Mathematics, a finite group is a group which has finitely many elements An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

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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 Mathematics, an n+1 cobordism is a Triple (WMN where W is an (n+1-dimensional Manifold, whose In Algebraic topology, a branch of Mathematics, a homotopy sphere is an n - Manifold homotopy equivalent to the n - sphere 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. In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not 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 Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, U Stephen Smale (born July 15, 1930) is an American Mathematician from Flint Michigan. Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St 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. In Mathematics, a parallelizable manifold M is a Smooth manifold of dimension n having Vector fields V 1 The structures of bPn+1 and the quotient

Θn/bPn+1

are described separately in the paper (Michel Kervaire & John Milnor 1963). Michel André Kervaire ( Częstochowa, Poland, 26 April, 1927 &ndash Geneva, Switzerland, 19 November,

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. In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. (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). In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other In Mathematics, the J -homomorphism is a mapping from the Homotopy groups of the Special orthogonal groups to the Homotopy groups of spheres 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). The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

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. In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other

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). In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric For a different notion of boundary related to Manifolds see that article Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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, . In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, . . , 28 gives all 28 possible smooth structures on the oriented 7-sphere.

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. Stephen Smale (born July 15, 1930) is an American Mathematician from Flint Michigan. (In contrast, in the PL setting, via radial extension the left-most map is onto: there are no PL-twisted spheres. Alexander's trick, also known as the Alexander trick, is a basic result in Geometric topology, named after J ) 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.

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:

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. )

See also

References

The Encyclopaedia of Mathematics is a large reference work in Mathematics.
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