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Flat Vector Bundle
In mathematics, a vector bundle is said to be ''flat'' if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. de Rham cohomology of a flat vector bundle Let \pi:E \to X denote a flat vector bundle, and \nabla : \Gamma(X, E) \to \Gamma\left(X, \Omega_X^1 \otimes E\right) be the covariant derivative associated to the flat connection on E. Let \Omega_X^* (E) = \Omega^*_X \otimes E denote the vector space (in fact a sheaf of modules over \mathcal O_X) of differential forms on ''X'' with values in ''E''. The covariant derivative defines a degree-1 endomorphism ''d'', the differential of \Omega_X^*(E), and the flatness condition is equivalent to the property d^2 = 0. In other words, the graded vector space \Omega_X^* (E) is a cochain complex. Its cohomology is called the de Rham cohomology of ''E'', or de Rham cohomology with coefficients twisted by the local coefficient system ''E''. Flat trivializations A trivialization of a flat vector bun ... [...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] |
Differential Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, th ... [...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] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''monodromy'' comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be ''single-valued'' as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called ''polydromy''. Definition Let be a connected and locally connected based topological space with base point , and let p: \tilde \to X be a covering with fiber F = p^(x). For a loop based at , denote a lift under the covering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian variety. Introduction The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety, of dimension ''g'', and hence, over the complex numbers, it is a complex torus. If ''p'' is a point of ''C'', then the curve ''C'' can be mapped to a subvariety of ''J'' with the given point ''p'' mapping to the identity of ''J'', and ''C'' generates ''J'' as a group. Construction for complex curves Over the complex numbers, the Jacobian variety can be realized as the quotient space ''V''/''L'', where ''V'' is the dual of the vector space of all global holomorphic differentials on ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its '' ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting shea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Poincaré Duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. Twisted Poincaré duality for de Rham cohomology Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle. This is the flat real line bundle denoted o(M), that is trivialized by coordinate charts of the manifold M, with transition maps the sign of the Jacobian determinant of the charts transition maps. As a flat line bundle, it has a de Rham cohomology, denoted by :H^* (M; \R^w) or H^* (M; o(M)). For ''M'' a compact manifold, the top degree cohomology is equipped with a so-called trace morphism :\theta\colon H^d (M; o(M)) \to \R, that is to be interpreted as integration on ''M'', ''i.e.'', evaluating against the fundamental class. Poincaré duality for different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Character
In algebraic topology, a branch of mathematics, an ''orientation character'' on a group \pi is a group homomorphism :\omega\colon \pi \to \left\. This notion is of particular significance in surgery theory. Motivation Given a manifold ''M'', one takes \pi=\pi_1 M (the fundamental group), and then \omega sends an element of \pi to -1 if and only if the class it represents is orientation-reversing. This map \omega is trivial if and only if ''M'' is orientable. The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving. Twisted group algebra The orientation character defines a twisted involution ( *-ring structure) on the group ring \mathbf pi/math>, by g \mapsto \omega(g)g^ (i.e., \pm g^, accordingly as g is orientation preserving or reversing). This is denoted \mathbf{Z} pi\omega. Examples *In real projective spaces, the orientation character evaluates triv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local System
In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. Definition Let ''X'' be a topological space. A local system (of abelian groups/modules/...) on ''X'' is a locally constant sheaf (of abelian groups/modules...) on ''X''. In other words, a sheaf \mathcal is a local system if every point has an open neighborhood U such that the restricted sheaf \mathcal, _U is isomorphic to the sheafification of some constant presheaf. Equivalent definitions Path-connected spaces If ''X'' is path-connected, a local system \mathcal of abelian groups has the same stalk ''L'' at eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-valued Differential Forms
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambda^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
