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Thom's First Isotopy Lemma
In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M \to N between smooth manifolds and S \subset M a closed Whitney stratified subset, if f, _S is proper and f, _A is a submersion for each stratum A of S, then f, _S is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when N = \mathbb. In that case, the lemma constructs an isotopy from the fiber f^(a) to f^(b); whence the name "isotopy lemma". The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth (not even C^1). On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic. The lemma is also valid for a more general stratified space such as a stratified space in the sense of Mather but still with the Whitney conditions (or some other conditions). The lemma is also valid for the stratification that satisfies Bekk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (connected) manifolds in each dimension separately: * In di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Conditions
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. A stratification of a topological space is a finite filtration by closed subsets ''F''''i'' , such that the difference between successive members ''F''''i'' and ''F''(''i'' − 1) of the filtration is either empty or a smooth submanifold of dimension ''i''. The connected components of the difference ''F''''i'' − ''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 R''n'' Let ''X'' and ''Y'' be two disjoint (locally closed) submanifolds of R''n'', of dimensions ''i'' and ''j''. * ''X'' and ''Y'' satisfy Whitney's condition A if whenever a sequence of points ''x''1, ''x''2, … in ''X'' converges to a point ''y'' in ''Y'', and the sequence of ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. Formal definitions Homotopy lifting property A mapping p \colon E \to B satisfies the homotopy lifting property for a space X if: * for every homotopy h \colon X \times , 1\to B and * for every mapping (also called lift) \tilde h_0 \colon X \to E lifting h, _ = h_0 (i.e. h_0 = p \circ \tilde h_0) there exists a (not necessarily unique) homotopy \tilde h \colon X \times , 1\to E lifting h (i.e. h = p \circ \tilde h) with \tilde h_0 = \tilde h, _. The following commutative diagram shows the situation:^ Fibration A fibration (also called Hurewicz fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all spaces X. The space B is called base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. 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 founder of catastrophe theory (later developed by Erik Christopher Zeeman). Life and career René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares''), was w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mather Stratified Space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum. Among the several ideals, Grothendieck's ''Esquisse d’un programme'' considers (or proposes) a stratified space with what he calls the tame topology. A stratified space in the sense of Mather Mather gives the following definition of a stratified space. A ''prestratification'' on a topological space ''X'' is a partition of ''X'' into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata ''A'', ''B' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom's Second Isotopy Lemma
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 René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w .... gives a sketch of the proof. gives a simplified proof. Like the first isotopy lemma, the lemma also holds for the stratification with Bekka's condition (C), which is weaker than Whitney's condition (B).§ 3 of Thom mapping Let f : M \to N be a smooth map between smooth manifolds and X, Y \subset M submanifolds such that f, _X, f, _Y both have differential of constant rank. Then Thom's condition (a_f) is said to hold if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tubular Neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood. In general, let ''S'' be a submanifold of a manifold ''M'', and let ''N'' be the normal bundle of ''S'' in ''M''. Here ''S'' plays the role of the curve and ''M'' the role of the plane containing the curve. Consider the natural map :i : N_0 \to S which establishes a bijective correspondence between the zero section N_0 of ''N'' and the su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehresmann's Fibration Theorem
In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f\colon M \rightarrow N, where M and N are smooth manifolds, is # a surjective submersion, and # a proper map (in particular, this condition is always satisfied if ''M'' is compact), then it is a locally trivial fibration. This is a foundational result in differential topology due to Charles Ehresmann, and has many variants. See also *Thom's first isotopy lemma References * * {{cite book, last1=Kolář, first1=Ivan, last2=Michor, first2=Peter W., last3=Slovák, first3=Jan, title=Natural operations in differential geometry, publisher=Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ..., location=Berlin, year=1993, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom–Mather Stratified Space
In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space ''X'' that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Hassler Whitney, Whitney. They were introduced by René Thom, who showed that every Whitney conditions , Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather (mathematician), John Mather in 1970, inspired by Thom's proof. Basic examples of Thom–Mather stratified spaces include manifolds with boundary (top dimension and codimension 1 boundary) and manifold with corners, manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex analytic varieties, or orb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tame Topology
In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program ''Esquisse d’un programme'' under the French name ''topologie modérée'' (moderate topology). It is a topology in which the theory of dévissage can be applied to stratified structures such as semialgebraic or semianalytic sets. Some authors consider an o-minimal structure to be a candidate for realizing tame topology in the real case. Some other authors have claimed that their theory of conically smooth stratified spaces can achieve tame topology. See also *Thom's first isotopy lemma In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M \to N between smooth manifolds and S \subset M a closed Whitney stratified subset, if f, _S is proper and f, _A is a submersion for eac ... References * External links *https://ncatlab.org/nlab/show/tame+topology Algebraic analysis Geometry education Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
