Whitney conditions

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In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.

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A stratification of a topological space is a finite filtration by closed subsets Fi , 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. 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.

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 1977 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 transverse to any countable family of submanifolds of M is always dense by Thom's transversality theorem. The density of the set of transverse 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 sets (due to René Thom) and subanalytic sets (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 Kraków, Poland, the Vietnamese mathematician Ta Lê Loi proved further that every definable set in an o-minimal structure can be given a Whitney stratification.[ citation needed ]

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In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that transversality is a generic property: any smooth map , may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold . Together with the Pontryagin–Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem.

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In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces is locally trivial when it is a Thom mapping. Like the first isotopy lemma, the lemma was introduced by René Thom.

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