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String Group
In topology, a branch of mathematics, a string group is an infinite-dimensional group \operatorname(n) introduced by as a 3-connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups0\rightarrow\rightarrow \operatorname(n)\rightarrow \operatorname(n)\rightarrow 0where K(\mathbb,2) is an Eilenberg–MacLane space and \operatorname(n) is a spin group. The string group is an entry in the Whitehead tower (dual to the notion of Postnikov tower) for the orthogonal group:\cdots\rightarrow \operatorname(n) \to \operatorname(n)\rightarrow \operatorname(n)\rightarrow \operatorname(n) \rightarrow \operatorname(n) It is obtained by killing the \pi_3 homotopy group for \operatorname(n), in the same way that \operatorname(n) is obtained fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids. Definition A 2-group is a monoidal category ''G'' in which every morphism is invertible and every object has a weak inverse. (Here, a ''weak inverse'' of an object ''x'' is an object ''y'' such that ''xy'' and ''yx'' are both isomorphic to the unit object.) Strict 2-groups Much of the literature focuses on ''strict 2-groups''. A strict 2-group is a ''strict'' monoidal category in which every morphism is invertible and every object has a strict inverse (so that ''xy'' and ''yx'' are actually equal to the unit object). A strict 2-group is a group object in a category of categories; as such, they are also called ''groupal categories''. Conversely, a strict 2-group is a category object in the category of groups; as such, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Bordism
String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian animated short * ''Strings'' (2004 film), a film directed by Anders Rønnow Klarlund * ''Strings'' (2011 film), an American dramatic thriller film * ''Strings'' (2012 film), a British film by Rob Savage * ''Bravetown'' (2015 film), an American drama film originally titled ''Strings'' * ''The String'' (2009), a French film Music Instruments * String (music), the flexible element that produces vibrations and sound in string instruments * String instrument, a musical instrument that produces sound through vibrating strings ** List of string instruments * String piano, a pianistic extended technique in which sound is produced by direct manipulation of the strings, rather than striking the piano's keys Types of groups * String band, musical ens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. History and motivation Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if S^1 acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning S^1-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerbe
In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (mathematician), Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of Deformation theory, deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf (agriculture), sheaf. Definitions Gerbes on a topological space A gerbe on a topological space S is a stack (mathematics), stack \mathcal of groupoids over S which is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-group (category Theory)
In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoà ng Xuân SÃnh was an in-depth study of 2-groups under the moniker 'gr-category'. The general definition of n-group is a matter of ongoing research. However, it is expected that every topological space will have a ''homotopy n-group'' at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group \pi_n, or the entire Postnikov tower for n=\infty. Examples Eilenberg-Maclane spaces One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces K(A,n) since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group G can be turned into an Eilenberg-Maclane space K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
