Whitney embedding theorem
Encyclopedia
In mathematics
, particularly in differential topology
, there are two Whitney embedding theorems:
with transverse
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. 
Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in 
. Since 
is simply-connected, 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 a family of immersions
of 
into 
which are approximately linear outside of the unit ball, but containing a single double point. For 
such an immersion is defined as 
with 
. Notice that if 
is considered as a map to 
i.e.: 
then the double point can be resolved to an embedding: 
. Notice 
and for 
then as a function of 
, 
is an embedding. Define 
. 
can similarly be resolved in 
, this process ultimately leads one to the definition: 
with 
for all 
. The key properties of 
is that it is an embedding except for the double-point 
. Moreover, for 
large, it is approximately the linear embedding 
.
in dimensions
, and the classification of smooth structure
s on discs (also in dimensions 5 and up).
This provides the foundation for surgery theory
, which classifies manifolds in dimension 5 and above.
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension
, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.
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 manifold extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties
for context.
-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 and the Klein bottle show. More generally, for 
we have 
, as the 
-dimensional real projective space
show. Whitney's result can be improved by showing that
unless 
is a power of 2. This is a result of Haefliger
–Hirsch
(
) and C.T.C. Wall (
); these authors used important preliminary results and particular cases proved by M. Hirsch, W. Massey
, S. Novikov and V. Rokhlin, see section 2 of this survey. At present the function
is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).
For example, the n-sphere always embeds in
– which is the best possible (closed n-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 A. Haefliger
-M. Hirsch
(
) and F. Fang (
); these authors used important preliminary results proved by J. Bo'echat-A. Haefliger, S. Donaldson
, M. Hirsch and W. Massey
. Haefliger proved that if
is a compact 
-dimensional 
-connected manifold, then 
embeds in 
provided 
.
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 
). See further generalizations.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, there are two Whitney embedding theorems:
- The strong Whitney embedding theorem states that any smoothDifferentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
m-dimensional manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
(required also to be HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and second-countable) can be smoothly embeddedEmbeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in EuclideanEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
-space, if m>0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaceReal projective spaceIn mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
s of dimension
cannot be embedded into Euclidean (
)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
- The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m>2n. Whitney similarly proved that such a map could be approximated by an immersion provided m>2n-1. This last result is sometimes called the weak Whitney immersion theorem.
A little about the proof
The general outline of the proof is to start with an immersion with transverseGlossary of differential geometry and topology
This is a glossary of terms specific to differential geometry and differential topology.The following two glossaries are closely related:*Glossary of general topology*Glossary of Riemannian and metric geometry.See also:*List of differential geometry topics...
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. . Since is simply-connected, 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).To introduce a local double point, Whitney created a family of immersions
of into which are approximately linear outside of the unit ball, but containing a single double point. For such an immersion is defined as with . Notice that if is considered as a map to i.e.: then the double point can be resolved to an embedding: . Notice and for then as a function of , is an embedding. Define . can similarly be resolved in , this process ultimately leads one to the definition: with for all . 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 Steve Smale to prove the h-cobordism theorem; from which follows the Poincaré conjecturePoincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
in dimensions
, and the classification of smooth structureSmooth 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....
s on discs (also in dimensions 5 and up).
This provides the foundation for surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
, which classifies manifolds in dimension 5 and above.
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension
, 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 WhitneyHassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
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 manifold extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties
History of manifolds and varieties
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In...
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 and the Klein bottle show. More generally, for we have , as the -dimensional real projective spaceReal projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
show. Whitney's result can be improved by showing that
unless is a power of 2. This is a result of HaefligerAndré Haefliger
André Haefliger is a Swiss mathematician who works primarily on topology.He studied mathematics in Lausanne. He received his PhD in 1958 from the University of Strasbourg under the supervision of Charles Ehresmann with "Structures feuilletées et cohomologie à valeurs dans un faisceau de...
–Hirsch
Morris Hirsch
Morris William Hirsch 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 Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...
(
) and C.T.C. Wall (); these authors used important preliminary results and particular cases proved by M. Hirsch, W. MasseyWilliam S. Massey
William Schumacher Massey is 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 couples, and wrote several textbooks, including Algebraic Topology .William Massey was...
, S. Novikov and V. Rokhlin, see section 2 of this survey. At present the function
is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).Restrictions on manifolds
One can strengthen the results by putting additional restrictions on the manifold.For example, the n-sphere always embeds in
– which is the best possible (closed n-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 A. HaefligerAndré Haefliger
André Haefliger is a Swiss mathematician who works primarily on topology.He studied mathematics in Lausanne. He received his PhD in 1958 from the University of Strasbourg under the supervision of Charles Ehresmann with "Structures feuilletées et cohomologie à valeurs dans un faisceau de...
-M. Hirsch
Morris Hirsch
Morris William Hirsch 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 Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...
(
) and F. Fang (); these authors used important preliminary results proved by J. Bo'echat-A. Haefliger, S. DonaldsonSimon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
, M. Hirsch and W. Massey
William S. Massey
William Schumacher Massey is 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 couples, and wrote several textbooks, including Algebraic Topology .William Massey was...
. 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 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
André Haefliger
André Haefliger is a Swiss mathematician who works primarily on topology.He studied mathematics in Lausanne. He received his PhD in 1958 from the University of Strasbourg under the supervision of Charles Ehresmann with "Structures feuilletées et cohomologie à valeurs dans un faisceau de...
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 ). See further generalizations.External links
See also
- Whitney immersion theorem
- Nash embedding theoremNash embedding theoremThe Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...