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L-theory
In mathematics, algebraic ''L''-theory is the ''K''-theory of quadratic forms; the term was coined by C. T. C. Wall, with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as "Hermitian ''K''-theory", is important in surgery theory. Definition One can define ''L''-groups for any ring with involution ''R'': the quadratic ''L''-groups L_*(R) (Wall) and the symmetric ''L''-groups L^*(R) (Mishchenko, Ranicki). Even dimension The even-dimensional ''L''-groups L_(R) are defined as the Witt groups of ε-quadratic forms over the ring ''R'' with \epsilon = (-1)^k. More precisely, ::L_(R) is the abelian group of equivalence classes psi/math> of non-degenerate ε-quadratic forms \psi \in Q_\epsilon(F) over R, where the underlying R-modules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic ε-quadratic forms: : psi= psi'\Longleftrightarrow n, n' \in _0: \psi \oplus H_(R)^n \cong \psi' \oplus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polarization Identities
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two Vector (geometry), vectors in terms of the Norm (mathematics), norm of a normed vector space. If a Norm (mathematics), norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. The norm associated with any inner product space satisfies the parallelogram law: \, x+y\, ^2 + \, x-y\, ^2 = 2\, x\, ^2 + 2\, y\, ^2. In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. Given a Normed vector space, normed space (H, \, \cdot\, ), the parallelogram law holds for \, \cdot\, if and only if there exists an inner product \langle \cdot, \cdot \rangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Group
In mathematics, a Witt group of a field (mathematics), field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear form, symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic (algebra), characteristic not equal to two. All vector spaces will be assumed to be finite-dimension (vector space), dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane (quadratic forms), hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p. 14 Each class is represented by the core form of a Witt decomposition.Lorenz (2008) p. 30 The Witt group of ''k'' is the abelian group ''W''(''k'') of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation correspo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Invariant
In geometric topology, the de Rham invariant is a mod 2 invariant of a (4''k''+1)-dimensional manifold, that is, an element of \mathbf/2 – either 0 or 1. It can be thought of as the simply-connected ''symmetric'' L-group L^, and thus analogous to the other invariants from L-theory: the signature, a 4''k''-dimensional invariant (either symmetric or quadratic, L^ \cong L_), and the Kervaire invariant, a (4''k''+2)-dimensional ''quadratic'' invariant L_. It is named for Swiss mathematician Georges de Rham, and used in surgery theory. Definition The de Rham invariant of a (4''k''+1)-dimensional manifold can be defined in various equivalent ways: * the rank of the 2-torsion in H_(M), as an integer mod 2; * the Stiefel–Whitney number w_2w_; * the (squared) Wu number, v_Sq^1v_, where v_ \in H^(M;Z_2) is the Wu class of the normal bundle of M and Sq^1 is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class In mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kervaire Invariant
In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected ''quadratic'' L-group L_, and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, L^ \cong L_), and the De Rham invariant, a (4k+1)-dimensional ''symmetric'' invariant L^. In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1. The Kervaire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arf Invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to , even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature (topology)
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold ''M'' of dimension divisible by four. This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem. Definition Given a connected and oriented manifold ''M'' of dimension 4''k'', the cup product gives rise to a quadratic form ''Q'' on the 'middle' real cohomology group :H^(M,\mathbf). The basic identity for the cup product :\alpha^p \smile \beta^q = (-1)^(\beta^q \smile \alpha^p) shows that with ''p'' = ''q'' = 2''k'' the product is symmetric. It takes values in :H^(M,\mathbf). If we assume also that ''M'' is compact, Poincaré duality identifies this with :H^(M,\mathbf) which can be identified with \mathbf. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on ''H''2''k''(''M'',''R''); and therefore to a quadratic form ''Q''. The form ''Q'' is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Novikov Conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants. The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture. Precise formulation of the conjecture Let G be a discrete group and BG its classifying space, which is an Eilenberg–MacLane space of type K(G,1), and therefore unique up to homotopy equivalence as a CW complex. Let :f\colon M\rightarrow BG be a continuous map from a closed oriented n-dimensional manifold M to BG, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singly Even
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization. *A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd. *A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others. Definitions The ancient Greek terms "even-times-even" ( grc, ἀρτιάκ ... [...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 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 Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Even
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization. *A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd. *A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others. Definitions The ancient Greek terms "even-times-even" ( grc, ἀρτιάκ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
