On the localization formula in equivariant cohomology

Pedroza, Andrés; Tu, Loring W.

doi:10.1016/j.topol.2005.10.013

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…There are many powerful theorems comparing the cohomology of a G-variety X with its equivariant cohomology ( [6] contains many examples), and the equivariant cohomology of X was already used to compute a Gysin map in [9].…”

Section: Introductionmentioning

confidence: 99%

Residues in group completions and the Cech cohomology of BG

Dewey¹

2016

Preprint

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Let G be a connected affine algebraic group over C, G → X be an open immersion of G-varieties, Z = X − G and i : Z → X be the inclusion. Let α ∈ H * (G, C) be primitive. We give a method to compute the image of α in H * (Z, i ! CX ), using a lift of α along the first edge map of the Čech spectral sequence for H * (BG, C). We apply it to the wonderful compactification of a centerless semisimple group G.

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Section: Introductionmentioning

confidence: 99%

Residues in group completions and the Cech cohomology of BG

Dewey¹

2016

Preprint

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“…This article computes the Gysin map of a fiber bundle with equivariantly formal fibers. In the companion article [21], we compute the Gysin map of a G-equivariant map for a compact connected Lie group G.…”

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confidence: 99%

Computing the Gysin Map Using Fixed Points

2017

Algebraic Geometry and Number Theory

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ABSTRACT. The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classical formulas to fiber bundles with equivariantly formal fibers.In enumerative algebraic geometry, to count the number of objects satisfying a set of conditions, one method is to represent the objects satisfying each condition by cycles in a parameter space M and then to compute the intersection of these cycles in M. When the parameter space M is a compact oriented manifold, by Poincaré duality, the intersection of cycles can be calculated as a product of classes in the rational cohomology ring. Sometimes, a cycle B in M is the image f (A) of a cycle A in another compact oriented manifold E under a map f : E → M. In this case the homology class [B] of B is the image f * [A] of the homology class of A under the induced map f * : H * (E) → H * (M) in homology, and the Poincaré dual η B of B is the image of the Poincaré dual η A of A under the map H * (E) → H * (M) in cohomology corresponding to the induced map f * in homology. This map in cohomology, also denoted by f * , is called the Gysin map, the Gysin homomorphism, or the pushforward map in cohomology. It is defined by the commutative diagramwhere e and m are the dimensions of E and M respectively and the vertical maps are the Poincaré duality isomorphisms. The calculation of the Gysin map for various flag bundles plays an important role in enumerative algebraic geometry, for example in determining the cohomology classes of degeneracy loci ([22], [19], [15], and [13, Ch. 14]). Other applications of the Gysin map, for example, to the computation of Thom polynomials associated to ThomBoardman singularities and to the computation of the dual cohomology classes of bundles of Schubert varieties, may be found in [12].

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Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Donten-Bury

Weber

2017

Transformation Groups

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Abstract. We study properties of the Hirzebruch class of quotient singularities C n /G, where G is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of G under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of G. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.

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On the localization formula in equivariant cohomology

Cited by 3 publications

References 15 publications

Residues in group completions and the Cech cohomology of BG

Residues in group completions and the Cech cohomology of BG

Computing the Gysin Map Using Fixed Points

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

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