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Splitting Principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, for example of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. Then the splitting principle can be quite useful. Statement One version of the splitting principle is captured in the following theorem. This theorem holds for complex vector bundles and cohomology with integer coefficients. It also holds for real vector bundles and cohomology with \mathbb_2 coefficients. In the complex case, the line bundles L_i or their first characteristic classes are called Chern roots. Another version of the splitting principle concerns real vector bundles and their complexifications: Consequences The fact that p^*\colon H^*(X)\rightarrow H^*(Y) is injective means that any equation which holds in H^*(Y) — for example, among ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Classes
In mathematics, in particular in algebraic topology, differential geometry and topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundle, complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten theory, Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact Space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracomp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Definition Let ''G'' be a topological group, and for a topological space X, write b_G(X) for the set of isomorphism classes of principal ''G''-bundles over X. This b_G is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f\colon X\to Y to the pullback operation f^*\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Polynomials
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every ''symmetric'' polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Splitting Principle
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry, homological algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Early life Bott was born in Budapest, Kingdom of Hungary (1920–1946), Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather in Bratislava, Czechoslovakia, now the capital of Slovakia. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Forces, Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
