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Volodin Space
In mathematics, more specifically in topology, the Volodin space X of a ring ''R'' is a subspace of the classifying space BGL(R) given by :X = \bigcup_ B(U_n(R)^\sigma) where U_n(R) \subset GL_n(R) is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and \sigma a permutation matrix thought of as an element in GL_n(R) and acting (superscript) by conjugation. The space is acyclic and the fundamental group \pi_1 X is the Steinberg group \operatorname(R) of ''R''. In fact, showed that ''X'' yields a model for Quillen's plus-construction BGL(R)/X \simeq BGL^+(R) in algebraic K-theory. Application An analogue of Volodin's space where GL(''R'') is replaced by the Lie algebra \mathfrak(R) was used by to prove that, after tensoring with Q, relative ''K''-theory K(''A'', ''I''), for a nilpotent ideal ''I'', is isomorphic to relative cyclic homology HC(''A'', ''I''). This theorem was a pioneering result in the area ... [...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] |
Plus Construction
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. Explicitly, if X is a based connected CW complex and P is a perfect normal subgroup of \pi_1(X) then a map f\colon X \to Y is called a +-construction relative to P if f induces an isomorphism on homology, and P is the kernel of \pi_1(X) \to \pi_1(Y).Charles Weibel, ''An introduction to algebraic K-theory'' IV, Definition 1.4.1 The plus construction was introduced by , and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex X, attach two-cells along loops in X whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells. The most common application of the plus construction is in algebraic K-theory. If R is a unital r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a "map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Method
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Homology
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology) and Alain Connes (cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. Contributors to the development of the theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg. Hints about definition The first definition of the cyclic homology of a ring ''A'' over a field of characteristic zero, denoted :''HC''''n''(''A'') or ''H''''n''λ(''A''), proceeded by the means of the following explicit c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinberg Group (K-theory)
In algebraic K-theory, a field of mathematics, the Steinberg group \operatorname(A) of a ring A is the universal central extension of the commutator subgroup of the stable general linear group of A . It is named after Robert Steinberg, and it is connected with lower K -groups, notably K_ and K_ . Definition Abstractly, given a ring A , the Steinberg group \operatorname(A) is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension). Presentation using generators and relations A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form (\lambda) := \mathbf + (\lambda) , where \mathbf is the identity matrix, (\lambda) is the matrix with \lambda in the (p,q) -entry and zeros elsewhere, and p \neq q — satisfy the following relations, called the Steinberg relations: : \begin e_(\lamb ... [...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] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Space
In mathematics, an acyclic space is a nonempty topological space ''X'' in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of ''X'' are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology, :\tilde_i(X)=0, \quad \forall i\ge -1. It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." The condition of acyclicity on a space ''X'' implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of ''X'' to the circle or to the higher spheres is null-homotopic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
