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Kervaire–Milnor Group
In mathematics, especially differential topology and cobordism theory, a Kervaire–Milnor group is an abelian group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation as inversion. It controls the existence of smooth structures on topological and piecewise linear (PL) manifolds. Concerning the related question of PL structures on topological manifolds, the obstruction is given by the Kirby–Siebenmann invariant, which is a lot easier to understand. In all but three and four dimensions, Kervaire–Milnor groups furthermore give the possible smooth structures on spheres, hence exotic spheres. They are named after the French mathematician Michel Kervaire and the American mathematician John Milnor, who first described them in 1962. (Their paper was originally only supposed to be the first part, but a second part was never published.) Definition An important property of spheres is their neutrality with respect ... [...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] |
Neutral Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromoll–Meyer Sphere
In mathematics, especially differential topology, the Gromoll–Meyer sphere is a special seven-dimensional exotic sphere with several unique properties. It is named after Detlef Gromoll and Wolfgang Meyer, who first described it in detail in 1974, although it was already found by John Milnor in 1956. Definition Brieskorn sphere In \mathbb^5 consider the complex variety: : a^2+b^2+c^2+d^3+e^5=0. A description of the Gromoll–Meyer sphere is the intersection of the above variety with a small sphere around the origin. Lie group biquotient The first symplectic group \operatorname(1) (isomorphic to \operatorname(2)) acts on the second symplectic group \operatorname(2) (isomorphic to \operatorname(5)) with the embedding \operatorname(1)\hookrightarrow\operatorname(2), q\mapsto\operatorname(q,q) and multiplication from the left as well as the embedding \operatorname(1)\hookrightarrow\operatorname(2), q\mapsto\operatorname(q,1) and multiplication from the right. A description of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brieskorn Manifold
In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by , is the intersection of a small sphere around the origin with the singular, complex hypersurface :x_1^+\cdots+x_n^=0 studied by . Brieskorn manifolds give examples of exotic sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of ...s, for example the Gromoll–Meyer sphere. References * * * This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds. * *{{Citation , last=Pham, first=Frédéric, authorlink=Frédéric Pham , title=Formules de Picard-Lefschetz généralisées et ramification des intégrales , mr=0195868 , year=1965 , journal= Bulletin de la Société Mathématique de France , issn=0037-9484 , volume=93 , pages=333–367, doi=10.24033/bsmf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936 in Rostock – 11 July 2013 in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution. Education Brieskorn was born in 1936 as the son of a mill construction engineer in East Prussia. He grew up in Freudenberg (Siegerland) and studied mathematics and physics at the Ludwig-Maximilians-Universität München and the Rheinische Friedrich-Wilhelms-Universität Bonn. In 1963 he received his doctorate at Bonn under Friedrich Hirzebruch with thesis ''Zur differentialtopologischen und analytischen Klassifizierung gewisser algebraischer Mannigfaltigkeiten'', followed by his habilitation in 1968. Career From 1969 until 1973 he was professor ordinarius at Georg-August-Universität Göttingen and from 1973 to 1975 at the Sonderforschungsbereich Theoretische Mathematik in Bonn (since 1980 called the Max-Planck-Institut für Mathematik). From 1975 until his retirement as professor emeritus in 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where G is the original group and N is the normal subgroup. This is read as '', where \text is short for modulo. (The notation should be interpreted w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or '' holes'', of a topological space. To define the ''n''th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never homeomorphic, but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. Introduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG \to BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is a path-connected topological space ''X'' such that the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by Function composition, composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrix, orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose invertible matrix, inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact group, compact. The orthogonal group in dimension has two connected component (topology), connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Equivalent
In topology, two continuous functions from one topological space to another are called homotopic (from and ) 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 parameter of ''H'' as time then ''H'' describes a ''continuous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms. Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in 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 ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
