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Ehresmann's 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 in ..., location=Berlin, year=1993, isbn=3-54 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Smooth Mapping
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion. Definition Let ''M'' and ''N'' be differentiable manifolds and f\colon M\to N be a differentiable map between them. The map is a submersion at a point p\in M if its differential :Df_p \colon T_p M \to T_N is a surjective linear map. In this case is called a regular point of the map , otherwise, is a critical point. A point q\in N is a regular value of if all points in the preimage f^(q) are regular points. A differentiable map that is a submersion at each point p\in M is called a submersion. Equivalently, is a submersion if its differential Df_p has constant rank equal to the dimension of . A word of warning: some authors use the term ''critical point'' to describe a point where the rank of the Jacobian matrix of at is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing definitions of a "proper function". Some authors call a function f : X \to Y between two topological spaces if the preimage of every compact set in Y is compact in X. Other authors call a map f if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff. Let f : X \to Y be a closed map, such that f^(y) is compact (in X) for all y \in Y. Let K be a compact subset of Y. It remains to show that f^(K) is compact. Let \left\ be an open cover of f^(K). Then for all k \in K this is also an open cover of f^(k). Since the latter is assumed to be compact, it has a finite subcover. In other words, for every k \in K, there exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Trivial
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold ... [...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] |
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory. Life Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber. Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34. He co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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