In an area of mathematics called
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, an exotic sphere is a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' that is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
but not
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
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 In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...
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
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
under connected sum, which is a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
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
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 28 under the operation of
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
.
Introduction
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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
between them, i.e. every point in ''X'' may be assigned to exactly one point in the unit ''n''-sphere in a
bicontinuous
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
(i.e. continuous and invertible with continuous inverse) manner. 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
. Similarly, an ''n''-cube of any radius can be continuously transformed to an ''n''-sphere.
In
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, the relevant notion of sameness is witnessed by a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
, which is a strict generalization of a homeomorphism. In particular, a more stringent condition is added requiring that the functions matching points in ''X'' with points in
should be
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
, that is they should have
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s 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 structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...
s 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
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
. 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 twist
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 ...
s. All homotopy ''n''-spheres are homeomorphic to the ''n''-sphere by the generalized
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
, proved by
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
in dimensions bigger than 4,
Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gene ...
in dimension 4, and
Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
in dimension 3. In dimension 3,
Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure (see
Moise's theorem
In geometric topology, a branch of mathematics, Moise's theorem, proved by Edwin E. Moise in , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure.
The analogue of Moise's theorem in ...
), so the monoid of smooth structures on the 3-sphere is trivial.
Parallelizable manifolds
The group
has a cyclic subgroup
:
represented by ''n''-spheres that bound
parallelizable manifold
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equi ...
s. The structures of
and the quotient
:
are described separately in the paper , which was influential in the development of
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
. In fact, these calculations can be formulated in a modern language in terms of the
surgery exact sequence
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension >4. The surgery structure set \mathcal (X) of a compact n-dimensional manifold X is ...
as indicated
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here
Television
* Here TV (form ...
.
The group
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 number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. 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
mod 4 is not of the form
. It follows from the now almost completely resolved
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphe ...
problem that it has order 2 for all ''n'' bigger than 126; the case
is still open. The order of
for
is
:
where ''B'' is the numerator of
, and
is a
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. (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
has a description in terms of
stable homotopy groups of spheres modulo the image of the
J-homomorphism
In mathematics, the ''J''-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .
Definition
Whitehead's original homomorphism is d ...
; it is either equal to the quotient or index 2. More precisely there is an injective map
:
where
is the ''n''th 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 number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. 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
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
2. The latter is the case if and only if there exists an ''n''-dimensional framed manifold with
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphe ...
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
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.
Order of Θn
The order of the group
is given in this table from (except that the entry for
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
, then
are
,
,
, and
. 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 that it is the last odd-dimensional sphere with this property – 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
be the unit ball in
, and let
be its
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
—a 3-sphere which we identify with the group of unit
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s. 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
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
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
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
of points in
satisfying
:
with a small sphere around the origin for
gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called
Brieskorn sphere In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by , is the intersection of a small sphere around the origin with the singular, complex hypersurface
:x_1^+\cdots+x_n^=0
studied by .
Brieskorn manifolds give examples ...
s.
Twisted spheres
Given an (orientation-preserving) diffeomorphism
, gluing the boundaries of two copies of the standard disk
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
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
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
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
of twisted spheres is always isomorphic to the group
. The notations are different because it was not known at first that they were the same for
or 4; for example, the case
is equivalent to the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
.
In 1970
Jean Cerf proved the
pseudoisotopy theorem which implies that
is the trivial group provided
, and so
provided
.
Applications
If ''M'' is a
piecewise linear manifold
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear ...
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:
*The group Θ
''n'' of h-cobordism classes of oriented homotopy ''n''-spheres.
*The group of h-cobordism classes of oriented ''n''-spheres.
*The group Γ
''n'' of twisted oriented ''n''-spheres.
*The homotopy group
''n''(PL/DIFF)
*If , the homotopy group
''n''(TOP/DIFF) (if this group has order 2; see
Kirby–Siebenmann invariant).
*The group of smooth structures of an oriented PL ''n''-sphere.
*If , the group of smooth structures of an oriented topological ''n''-sphere.
*If , the group of components of the group of all orientation-preserving diffeomorphisms of ''S''
''n''−1.
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, , , , , , , , .
See also
*
Milnor's sphere
In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of (n-1)-connected manifolds of dimension 2n (since n-connected 2n-manifolds are homeomorphic to sph ...
*
Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
*
Clutching construction In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Definition
Consider the sphere S^n as the union of the upper and lower hemispheres D^n_+ and D^n_- alon ...
*
Exotic R4
*
Cerf theory
*
Seven-dimensional space
In mathematics, a sequence of ''n'' real numbers can be understood as a location in ''n''-dimensional space. When ''n'' = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any n ...
References
*
*
*
*
*
*
*
*
*
*
*
*
*
* This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
* – This paper describes the structure of the group of smooth structures on an ''n''-sphere for ''n'' > 4. The promised paper "Groups of Homotopy Spheres: II" never appeared, but Levine's lecture notes contain the material which it might have been expected to contain.
*
*
*
*
*
*.
*
*
*
*.
* .
*{{springer, title=Milnor sphere, id=M/m063800, first=Yuli B., last=Rudyak
External links
Exotic sphereson the Manifold Atlas
on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres.
Video from a presentation b
Niles Johnsonat th
Second Abel conferencein honor of
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
.
The Gluck constructionon the Manifold Atlas
Differential topology
Differential structures
Surgery theory
Spheres