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Equivariant Stable Homotopy Theory
In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. The field has become more active recently because of its connection to algebraic K-theory. See also * Equivariant K-theory *G-spectrum In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group. Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set X^. There is always :X^G \to X^, a ma ... (spectrum with an action of an (appropriate) group ''G'') References External linksCreating Equivariant Stable Homotopy Theory Homotopy theory {{topology-stub ... [...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] |
Equivariant Topology
In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps f: X \to Y, and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry" in both its domain and target space. The notion of symmetry is usually captured by considering a group action of a group G on X and Y and requiring that f is equivariant under this action, so that f(g\cdot x) = g \cdot f(x) for all x \in X, a property usually denoted by f: X \to_ Y. Heuristically speaking, standard topology views two spaces as equivalent "up to deformation," while equivariant topology considers spaces equivalent up to deformation so long as it pays attention to any symmetry possessed by both spaces. A famous theorem of equivariant topology is the Borsuk–Ulam theorem, which asserts that every \mathbf_2-equivariant map f: S^n \to \mathbb R^n necessarily vanis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representable functor, representing a Cohomology#Generalized cohomology theories, generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left[X, E^k\right].Note there are several different category (mathematics), categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X, the homotopy groups \pi_(\Sigma^n X) stabilize for n sufficiently large. In particular, the homotopy groups of spheres \pi_(S^n) stabilize for n\ge k + 2. For example, :\langle \text_\rangle = \Z = \pi_1(S^1)\cong \pi_2(S^2)\cong \pi_3(S^3)\cong\cdots :\langle \eta \rangle = \Z = \pi_3(S^2)\to \pi_4(S^3)\cong \pi_5(S^4)\cong\cdots In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that \pi_n(S^n)\cong \Z. In the second example the Hopf map, \eta, is mapped to its suspension \Sigma\eta, which generates \pi_4(S^3)\cong \Z/2. One of the most i ... [...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] |
Equivariant Algebraic K-theory
In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category \operatorname^G(X) of equivariant coherent sheaves on an algebraic scheme ''X'' with action of a linear algebraic group ''G'', via Quillen's Q-construction; thus, by definition, :K_i^G(X) = \pi_i(B^+ \operatorname^G(X)). In particular, K_0^G(C) is the Grothendieck group of \operatorname^G(X). The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K_i^G(X) may be defined as the K_i of the category of coherent sheaves on the quotient stack /G/math>. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.) A version of the Lefschetz fixed-point theorem holds in the setting of equivariant (algebraic) K-theory. Fundamental theorems Let ''X'' be an equivariant algebraic scheme. Examples One of the fundamental examples of equivariant K-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-spectrum
In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group. Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set X^. There is always :X^G \to X^, a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, X^ is the mapping spectrum F(BG_+, X)^G.) Example: \mathbb/2 acts on the complex ''K''-theory ''KU'' by taking the conjugate bundle of a complex vector bundle. Then KU^ = KO, the real ''K''-theory. The cofiber of X_ \to X^ is called the Tate spectrum of ''X''. ''G''-Galois extension in the sense of Rognes This notion is due to J. Rognes . Let ''A'' be an E∞-ring with an action of a finite group ''G'' and ''B'' = ''A''''hG'' its invariant subring. Then ''B'' → ''A'' (the map of ''B''-algebras in E∞-sense) is said to be a ''G-Galois extension'' if the natural map :A \otimes_B A \to \prod_ A (which generalizes x \otimes y \mapsto (g(x) y) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
