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Localization Formula For Equivariant Cohomology
In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form \alpha on an orbifold ''M'' with a torus action and for a sufficient small \xi in the Lie algebra of the torus ''T'', : \int_M \alpha(\xi) = \sum_F \int_F where the sum runs over all connected components ''F'' of the set of fixed points M^T, d_M is the orbifold multiplicity of ''M'' (which is one if ''M'' is a manifold) and e_T(F) is the equivariant Euler form of the normal bundle of ''F''. The formula allows one to compute the equivariant cohomology ring of the orbifold ''M'' (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. No analog of such results holds in the non-equivariant cohomology. One important consequence of the formula is the Duistermaat–Heckman theorem, which states: supposing there is a Hamiltonian circle action (for simplicity) on a compact symplect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Differential Form
In differential geometry, an equivariant differential form on a manifold ''M'' acted upon by a Lie group ''G'' is a polynomial map :\alpha: \mathfrak \to \Omega^*(M) from the Lie algebra \mathfrak = \operatorname(G) to the space of differential forms on ''M'' that are equivariant; i.e., :\alpha(\operatorname(g)X) = g\alpha(X). In other words, an equivariant differential form is an invariant element ofProof: with V = \Omega^*(M), we have: \operatorname_G(\mathfrak, V) = \operatorname(\mathfrak, V)^G = (\operatorname(\mathfrak, \mathbb)\otimes V)^G. Note \mathbbmathfrak/math> is the ring of polynomials in linear functionals of \mathfrak; see ring of polynomial functions. See also https://math.stackexchange.com/q/101453 for M. Emerton's comment. :\mathbbmathfrak\otimes \Omega^*(M) = \operatorname(\mathfrak^*) \otimes \Omega^*(M). For an equivariant differential form \alpha, the equivariant exterior derivative d_\mathfrak \alpha of \alpha is defined by :(d_\mathfrak \alpha)(X) = d(\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties). Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus ''T'' is acting on a finite-dimensional vector space ''V'', then there is a direct sum decomposition: :V = \bigoplus_ V_ where *\chi: T \to \mathbb_m is a group homomorphism, a character of ''T''. *V_ = \, ''T''-invariant subspace called the weight subspace of weight \chi. The decomposition exists because the linear action determines (and is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold Multiplicity
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article E is an oriented, real vector bundle of rank r over a base space X. Formal definition The Euler class e(E) is an element of the integral cohomology group :H^r(X; \mathbf), constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology :H^r(\mathbf^, \mathbf^ \setminus \; \mathbf)\cong \tilde^(S^;\mathbf)\cong \mathbf of each fiber \mathbf^ relative to the complement \mathbf^ \setminus \ of zero. From the Thom isomorphism, this induces an orientation class :u \in H^r(E, E \setminus E_0; \mathbf) in the cohomology of E relative to the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Cohomology Ring
In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^*(X;A) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Stack
A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence. Differentiable stacks are particularly useful to handle spaces with singularities (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory. Definition Definition 1 (via groupoid fibrations) Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category \mathcal together with a functor \pi: \mathcal \to \mathrm to the category of differentiable manifolds such that # \mathcal is a fibred category, i.e. for any object u of \mathcal and any arrow V \to U of \mathrm there is an arrow v \to u lying over V \to U; # for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group Action
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition and first properties Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot x be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map \sigma is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism G \to \mathrm(M). A smooth manifold endowed with a Lie group action is also called a ''G''-manifold. The fact that the action map \sigma is smooth has a couple of immediate consequences: * the stabilizers G_x \subseteq G of the group action are closed, thus are Lie subgroups of ''G'' * the orbits G \cdot x \subseteq M of the group action are immersed submanifolds. Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action. Examples For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
