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Equivariant map
About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.
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TL;DR: In this article, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring H T * ( X ) can be described by combinatorial data obtained from its orbit decomposition.
109 citations
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TL;DR: Locally smooth S'-actions on simply connected 4-manifolds are studied in this paper in terms of their weighted orbit spaces, and an equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected S '-action.
Abstract: Locally smooth S'-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with S '-action. This is used to show that a simply connected 4-manifold which admits a locally smooth 5'-action must be homotopy equivalent to a connected sum of copies of S4, CP2, CP2, and S2 x S2.
109 citations
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TL;DR: In this article, the authors extend the notion of test categories with respect to some localizations of the homotopy category of CW-complexes, and prove two conjectures made by Grothendieck: any category of presheaves on a test category is canonically endowed with a Quillen closed model category structure.
Abstract: Grothendieck introduced in Pursuing Stacks the notion of test category These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes A well known example is the category of simplices (the corresponding presheaves are then simplicial sets) Moreover, Grothendieck defined the notion of basic localizer which gives an axiomatic approach to the homotopy theory of small categories, and gives a natural setting to extend the notion of test category with respect some localizations of the homotopy category of CW-complexes This text can be seen as a sequel of Grothendieck's homotopy theory We prove in particular two conjectures made by Grothendieck: any category of presheaves on a test category is canonically endowed with a Quillen closed model category structure, and the smallest basic localizer defines the homotopy theory of CW-complexes Moreover, we show how a local version of the theory allows to consider in a unified setting the equivariant homotopy theory as well The realization of this program goes through the construction and the study of model category structures on any category of presheaves on an abstract small category, as well as the study of the homotopy theory of small categories following and completing the contributions of Quillen, Thomason and Grothendieck
108 citations
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Abstract: The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group $G$. For a $G$-space $X$, this theorem gives an isomorphism between a completion of the equivariant Grothendieck group and a completion of equivariant equivariant Chow groups.
The key to proving this isomorphism is a geometric description of completions of the equivariant Grothendieck group. Besides Riemann-Roch, this result has some purely $K$-theoretic applications. In particular, we prove a conjecture of K\"ock (in the case of regular schemes) and extend to arbitrary characteristic a result of Segal on representation rings.
107 citations
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TL;DR: A geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries is given, which yields a class of observers such that the error equation is autonomous.
Abstract: In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.
107 citations