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In mathematics, an 'exotic sphere' is a differentiable manifold, ''M'', that is homeomorphic to the standard Euclidean ''n''-sphere, but not diffeomorphic. That means that ''M'' is a sphere from a topological point of view, but not from the point of view of its differential structure. Thus, if ''M'' has dimension ''n'', there is a homeomorphism
:''h'' : ''M'' → ''S''^''n'',
but no such ''h'' is a diffeomorphism.

Contents

History

The number of exotic spheres in a given dimension

Explicit examples

Twisted spheres

Gluck twists

See also

References

History

The first exotic spheres were constructed by John Milnor in the case ''n'' = 7.^[1] They were ''S''³-bundles over ''S''⁴. This type of exotic sphere is called a 'Milnor sphere'. Later techniques based on surgery theory enabled calculations of the numbers of distinct exotic spheres, in any given dimension. For dimension 7, there are 28 oriented exotic spheres, and 15 if you omit orientation.
In any dimension the classes of (oriented) exotic spheres form an abelian group under connected sum.^[2] In the seven-dimensional case, this group is cyclic of order 28.

The number of exotic spheres in a given dimension

The formula of Michel Kervaire and John Milnor^[3] for the order of the cyclic group of diffeomorphism classes of exotic (4''n'' − 1)-spheres which bound parallelizable manifolds for ''n'' ≥ 2 involves Bernoulli numbers. If ''B'' is the numerator of ''B''_4''n''/''n'', then
:$2^\{2n-2\}(1-2^\{2n-1\})B\; ,!$
is the number of such exotic spheres. (The formula in the topological literature differs because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)
The quotient group has a description in terms of homotopy theory (the J-homomorphism).
The order of the group of smooth structures on an oriented sphere in ''n'' dimensions is given in this table .
:

Dimension	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Structures	1	1	1	?	1	1	28	2	8	6	992	1	3	2	16256	2	16	16

In dimension 4 almost nothing is known about the group of smooth structures, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite. In dimensions other than 4 the group is finite (and abelian).

Explicit examples

Take two copies of ''B''⁴×''S''³, each with boundary
''S''³×''S''³, and glue them together by identifying (''a'',''b'') in the boundary with
(''a'', ''a''²''ba''⁻¹), (where we identify each ''S''³ with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to ''S''⁷, but is not diffeomorphic to ''S''⁷.
As shown by Egbert Brieskorn, the intersection of the complex manifold of points in 'C'⁵ satisfying
:''a''² + ''b''² + ''c''² + ''d''³ + ''e''^{6''k'' − 1} = 0
with a small sphere around the origin for ''k'' = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere.^[4]

Twisted spheres

Given an (orientation-preserving) diffeomorphism ''f'': ''S''^''n''−1→''S''^''n''−1, gluing the boundaries of two copies of the standard disk ''D''^''n'' 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.^[5] The term originated with John Milnor.^[6] (This should not be confused with the clutching construction, which glues fiber bundles.)
Setting $Gamma\_n$ to be the group of twisted ''n-''spheres (under connect sum), one obtains the exact sequence
:$pi\_0,\; ext\{Diff\}^+(D^n)\; o\; pi\_0,\; ext\{Diff\}^+(S^\{n-1\})\; o\; Gamma\_n\; o\; 0\; ,!$
For ''n'' > 4, every exotic sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale. (In contrast, in the PL setting, via radial extension the left-most map is onto: there are no PL-twisted spheres.)

Gluck twists

In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. Some candidates for such structures are given by 'Gluck twists'.^[7] These 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. But in most cases it is unknown whether or not the result is diffeomorphic to ''S''⁴. (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 ''S''⁴, but there are plently of other ways to knot a 2-sphere in ''S''⁴.)

See also

★ Exotic R⁴

★ Differential structure

References

1. John Milnor, ''On manifolds homeomorphic to the 7-sphere'', Annals of Mathematics, 64 (1956), no. 2, 399–405. This gives the first examples of exotic spheres.
2.
John W. Milnor, ''Sommes de variétes différentiables et structures différentiables des sphères'', Bull. Soc. Math. France 87 (1959), 439–444.
3. Michel A. Kervaire and John W. Milnor, ''Groups of homotopy spheres: I'', Annals of Mathematics, 77 (1963), no. 3, 504–537. This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4. Sadly, the promised paper Groups of Homotopy Spheres: II never appeared.
4. Friedrich Hirzebruch and Karl Heinz Mayer, ''O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten'', Lecture Notes in Mathematics, No. 57, Springer-Verlag, Berlin-New York, 1968. This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
5. John Milnor, ''Differentiable structures on spheres'', American Journal of Mathematics, 81 (1959), no. 4, 962–972.
6. John Milnor, "Fifty Years Ago: Topology of Manifolds in the 50's and 60's", June 27, 2006. More details can be found in Milnor's "Lectures on Differentiable Structures".
7. Herman Gluck, ''The embedding of two-spheres in the four-sphere'', Transactions of the American Mathematical Society 104 (1962), 308–333.