In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly 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 ...
, there are two Whitney embedding theorems, named after
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
:
*The strong Whitney embedding theorem states that any
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 ...
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(required also to be
Hausdorff and
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
) can be
smoothly embedded in the
real -space (), if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
s of dimension cannot be embedded into real -space if is a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
(as can be seen from a
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
argument, also due to Whitney).
*The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
provided . This last result is sometimes called the
Whitney immersion theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, an ...
.
A little about the proof
The general outline of the proof is to start with an immersion with
transverse
Transverse may refer to:
*Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle
*Transverse flute, a flute that is held horizontally
* Transverse force (or ''Euler force''), the tangen ...
self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If has boundary, one can remove the self-intersections simply by isotoping into itself (the isotopy being in the domain of ), to a submanifold of that does not contain the double-points. Thus, we are quickly led to the case where has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point.
![whitneytrickstep1](https://upload.wikimedia.org/wikipedia/commons/b/b9/Whitneytrickstep1.svg)
Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in . Since is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
, one can assume this path bounds a disc, and provided one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in such that it intersects the image of only in its boundary. Whitney then uses the disc to create a
1-parameter family of immersions, in effect pushing across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured). This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick.
To introduce a local double point, Whitney created immersions which are approximately linear outside of the unit ball, but containing a single double point. For such an immersion is given by
:
Notice that if is considered as a map to like so:
:
then the double point can be resolved to an embedding:
:
Notice and for then as a function of , is an embedding.
For higher dimensions ''m'', there are that can be similarly resolved in . For an embedding into , for example, define
:
This process ultimately leads one to the definition:
:
where
:
The key properties of is that it is an embedding except for the double-point . Moreover, for large, it is approximately the linear embedding .
Eventual consequences of the Whitney trick
The Whitney trick was used 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 ...
to prove the
''h''-cobordism theorem; from which follows 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 dimensions , and the classification 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 discs (also in dimensions 5 and up). This provides the foundation for
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 ...
, which classifies manifolds in dimension 5 and above.
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension ≥ 5, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.
History
The occasion of the proof by
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the ''manifold concept'' precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifolds extrinsically defined as submanifolds of Euclidean space. See also the
history of manifolds and varieties
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surface (topology), surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additio ...
for context.
Sharper results
Although every -manifold embeds in , one can frequently do better. Let denote the smallest integer so that all compact connected -manifolds embed in . Whitney's strong embedding theorem states that . For we have , as the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
and the
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
show. More generally, for we have , as the -dimensional
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
show. Whitney's result can be improved to unless is a power of 2. This is a result of
André Haefliger
André Haefliger (born 22 May 1929 in Nyon, Switzerland) is a Swiss mathematician who works primarily on topology.
Education and career
Haefliger went to school in Nyon and then attended his final years at Collège Calvin in Geneva. He studied ...
and
Morris Hirsch
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley.
A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of ...
(for ) and
C. T. C. Wall
Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where ...
(for ); these authors used important preliminary results and particular cases proved by Hirsch,
William S. Massey
William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact ...
,
Sergey Novikov and
Vladimir Rokhlin.
[See section 2 of Skopenkov (2008)] At present the function is not known in closed-form for all integers (compare to the
Whitney immersion theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, an ...
, where the analogous number is known).
Restrictions on manifolds
One can strengthen the results by putting additional restrictions on the manifold. For example, the
-sphere always embeds in – which is the best possible (closed -manifolds cannot embed in ). Any compact ''orientable'' surface and any compact surface ''with non-empty boundary'' embeds in , though any ''closed non-orientable'' surface needs .
If is a compact orientable -dimensional manifold, then embeds in (for not a power of 2 the orientability condition is superfluous). For a power of 2 this is a result of
André Haefliger
André Haefliger (born 22 May 1929 in Nyon, Switzerland) is a Swiss mathematician who works primarily on topology.
Education and career
Haefliger went to school in Nyon and then attended his final years at Collège Calvin in Geneva. He studied ...
and
Morris Hirsch
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley.
A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of ...
(for ), and Fuquan Fang (for ); these authors used important preliminary results proved by Jacques Boéchat and Haefliger,
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He i ...
, Hirsch and
William S. Massey
William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact ...
.
[ Haefliger proved that if is a compact -dimensional -connected manifold, then embeds in provided .][
]
Isotopy versions
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into R4 are isotopic. This is proved using general position, which also allows to show that any two embeddings of an -manifold into are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
Wu proved that for , any two embeddings of an -manifold into are isotopic. This result is an isotopy version of the strong Whitney embedding theorem.
As an isotopy version of his embedding result, Haefliger proved that if is a compact -dimensional -connected manifold, then any two embeddings of into are isotopic provided . The dimension restriction is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in (and, more generally, -spheres in ). Se
further generalizations
See also
* Representation theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
Examples
Algebra
* Cayley's theorem states that every group i ...
* Whitney immersion theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, an ...
* Nash embedding theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
* Takens's theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the prope ...
* Nonlinear dimensionality reduction
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-d ...
Notes
References
*
*
*
*
External links
Classification of embeddings
{{DEFAULTSORT:Whitney Embedding Theorem
Theorems in differential topology