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Whitney Immersion Theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, and a (not necessarily one-to-one) immersion in (2m-1)-space. Similarly, every smooth m-dimensional manifold can be immersed in the 2m-1-dimensional sphere (this removes the m>1 constraint). The weak version, for 2m+1, is due to transversality (general position, dimension counting): two ''m''-dimensional manifolds in \mathbf^ intersect generically in a 0-dimensional space. Further results William S. Massey went on to prove that every ''n''-dimensional manifold is cobordant to a manifold that immerses in S^ where a(n) is the number of 1's that appear in the binary expansion of n. In the same paper, Massey proved that for every ''n'' there is manifold (which happens to be a product of real projective spaces) that does not immerse in S^. ... [...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] |
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory. Biography Life Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge. His mother, A. Josepha Newcomb Whitney, was an artist and active in politics. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon Eben Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist and Sanskrit scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (1835-1909), a Steeves desce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of open subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R''n'') with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis. Properties Second-countability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at every point ''p'' of ''M'' (where ''TpX'' denotes the tangent space of a manifold ''X'' at a point ''p'' in ''X''). Equivalently, ''f'' is an immersion if its derivative has constant rank equal to the dimension of ''M'': :\operatorname\,D_p f = \dim M. The function ''f'' itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that ''M'' is diffeomorphic to its image in ''N''. An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of ''x'' such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversality (mathematics)
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection. Definition Two submanifolds of a given finite-dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point. Manifolds that do not intersect are vacuously transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the ambient space), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Counting
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William S
William is a male given name of Germanic origin.Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 276. It became very popular in the English language after the Norman conquest of England in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will, Wills, Willy, Willie, Bill, and Billy. A common Irish form is Liam. Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie or the play ''Douglas''). Female forms are Willa, Willemina, Wilma and Wilhelmina. Etymology William is related to the given name ''Wilhelm'' (cf. Proto-Germanic ᚹᛁᛚᛃᚨᚺᛖᛚᛗᚨᛉ, ''*Wiljahelmaz'' > German ''Wilhelm'' and Old Norse ᚢᛁᛚᛋᛅᚼᛅᛚᛘᛅᛋ, ''Vilhjálmr''). By regular sound changes, the native, inherited English form of the name shoul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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