Whitney conditions
Encyclopedia
In differential topology
, a branch of mathematics
, the Whitney conditions are conditions on a pair of submanifold
s of a manifold
introduced by Hassler Whitney
in 1965. A finite filtration by closed subsets Fi of a smooth manifold such that the difference between successive members Fi and F(i − 1) of the filtration is either empty or a smooth submanifold of dimension i, is called a stratification. The connected components of the difference Fi − F(i − 1) are the strata of dimension i. A stratification is called a Whitney stratification if all pairs of strata satisfy the Whitney conditions A and B, as defined below.
John Mather
first pointed out that Whitney's condition B implies Whitney's condition A in the notes of his lectures at Harvard in 1970, which have been widely distributed. He also defined the notion of Thom-Mather stratified space, and proved that every Whitney stratification is a Thom-Mather stratified space and hence is a topologically stratified space
. Another approach to this fundamental result was given earlier by René Thom
in 1969.
David Trotman
showed in his 1978 Warwick thesis that a stratification of a closed subset in a smooth manifold M satisfies Whitney's condition A if and only if the subspace of the space of smooth mappings from a smooth manifold N into M consisting of all those maps which are transverse to all of the strata of the stratification, is open (using the Whitney, or strong, topology). The subspace of mappings transversal to any countable family of submanifolds of M is always dense by Thom's transversality theorem. The density of the set of transversal mappings is often interpreted by saying that transversality is a 'generic' property for smooth mappings, while the openness is often interpreted by saying that the property is 'stable'.
The reason that Whitney conditions have become so widely used is because of Whitney's 1965 theorem that every algebraic variety, or indeed analytic variety, admits a Whitney stratification, i.e. admits a partition into smooth submanifolds satisfying the Whitney conditions. More general singular spaces can be given Whitney stratifications, such as semialgebraic set
s (due to René Thom
) and subanalytic set
s (due to Heisuke Hironaka
). This has led to their use in engineering, control theory and robotics. In a thesis under the direction of Wieslaw Pawlucki at the Jagellonian University in Krakow, Poland, the Vietnamese mathematician Ta Lê Loi proved further that every definable set in an o-minimal structure can be given a Whitney stratification.
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 :...
, a branch of mathematics
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...
, the Whitney conditions are conditions on a pair of submanifold
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...
s of a manifold
Manifold
In 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....
introduced by Hassler Whitney
Hassler 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:...
in 1965. A finite filtration by closed subsets Fi of a smooth manifold such that the difference between successive members Fi and F(i − 1) of the filtration is either empty or a smooth submanifold of dimension i, is called a stratification. The connected components of the difference Fi − F(i − 1) are the strata of dimension i. A stratification is called a Whitney stratification if all pairs of strata satisfy the Whitney conditions A and B, as defined below.
The Whitney conditions in Rn
Let X and Y be two disjoint locally closed submanifolds of Rn, of dimensions i and j.- X and Y satisfy Whitney's condition A if whenever a sequence of points x1, x2, … in X converges to a point y in Y, and the sequence of tangent i-planes Tm to X at the points xm converges to an i-plane T as m tends to infinity, then T contains the tangent j-plane to Y at y.
- X and Y satisfy Whitney's condition B if for each sequence x1, x2, … of points in X and each sequence y1, y2, … of points in Y, both converging to the same point y in Y, such that the sequence of secant lines Lm between xm and ym converges to a line L as m tends to infinity, and the sequence of tangent i-planes Tm to X at the points xm converges to an i-plane T as m tends to infinity, then L is contained in T.
John Mather
John Mather
John Norman Mather is a mathematician at Princeton University known for his work on singularity theory and Hamiltonian dynamics...
first pointed out that Whitney's condition B implies Whitney's condition A in the notes of his lectures at Harvard in 1970, which have been widely distributed. He also defined the notion of Thom-Mather stratified space, and proved that every Whitney stratification is a Thom-Mather stratified space and hence is a topologically stratified space
Topologically stratified space
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
. Another approach to this fundamental result was given earlier by René Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
in 1969.
David Trotman
David Trotman
David John Angelo Trotman is a mathematician, with dual British and French nationality. He was born on September 27, 1951 in Plymouth, Devon, England, a grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge...
showed in his 1978 Warwick thesis that a stratification of a closed subset in a smooth manifold M satisfies Whitney's condition A if and only if the subspace of the space of smooth mappings from a smooth manifold N into M consisting of all those maps which are transverse to all of the strata of the stratification, is open (using the Whitney, or strong, topology). The subspace of mappings transversal to any countable family of submanifolds of M is always dense by Thom's transversality theorem. The density of the set of transversal mappings is often interpreted by saying that transversality is a 'generic' property for smooth mappings, while the openness is often interpreted by saying that the property is 'stable'.
The reason that Whitney conditions have become so widely used is because of Whitney's 1965 theorem that every algebraic variety, or indeed analytic variety, admits a Whitney stratification, i.e. admits a partition into smooth submanifolds satisfying the Whitney conditions. More general singular spaces can be given Whitney stratifications, such as semialgebraic set
Semialgebraic set
In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
s (due to René Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
) and subanalytic set
Subanalytic set
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points defined in a way broader than for semianalytic sets...
s (due to Heisuke Hironaka
Heisuke Hironaka
is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....
). This has led to their use in engineering, control theory and robotics. In a thesis under the direction of Wieslaw Pawlucki at the Jagellonian University in Krakow, Poland, the Vietnamese mathematician Ta Lê Loi proved further that every definable set in an o-minimal structure can be given a Whitney stratification.
See also
- Whitney stratified space
- Thom-Mather stratified space
- Topologically stratified spaceTopologically stratified spaceIn topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...