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Koszul duality and equivariant cohomology for tori  [PDF]
Matthias Franz
Mathematics , 2003, DOI: 10.1155/S1073792803206103
Abstract: Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions about the spaces are made and that the coefficient ring may be arbitrary. This gives in particular a Cartan-type model for the equivariant cohomology of a T-space with arbitrary coefficients. Our method works for intersection homology as well.
Koszul duality and equivariant cohomology  [PDF]
Matthias Franz
Mathematics , 2003,
Abstract: Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between G-spaces and spaces over BG to the Koszul duality between modules up to homotopy over H(G) and H^*(BG). This gives in particular a Cartan-type model for the equivariant cohomology of a G-space. As another corollary, we obtain a multiplicative quasi-isomorphism C^*(BG) -> H^*(BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of A-infinity algebras provided that it is free over R and its homology an exterior algebra.
Relative homological algebra, equivariant de Rham theory, and Koszul duality  [PDF]
Johannes Huebschmann
Mathematics , 2004,
Abstract: Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham cohomology in terms of a suitable relative differential graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning of "relative" is made precise via the dual standard construction associated with the monad involving the aforementioned functor and the associated forgetful functor. The corresponding infinitesimal equivariant cohomology is the relative differential Ext over Cg relative to g. The functor under discussion decomposes into two functors, the functor which determines differentiable cohomology in the sense of Hochschild-Mostow and the functor which determines the infinitesimal equivariant theory, suitably interpreted. This functor decomposition, in turn, entails an extension of a Decomposition Lemma due to Bott. Appropriate models for the differential graded Ext involving a comparison between a suitably defined simplicial Weil coalgebra and the Weil coalgebra dual to the familiar ordinary Weil algebra yield small models for equivariant de Rham cohomology including the standard Weil and Cartan models for the special case where the group G is compact and connected. Koszul duality in de Rham theory results from these considerations in a straightforward manner.
Equivariant ordinary homology and cohomology  [PDF]
Steven R. Costenoble,Stefan Waner
Mathematics , 2003,
Abstract: Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of RO(G)-graded ordinary homology and cohomology, collecting scattered results and filling in gaps in the literature. In particular, we give details on change of groups and products that do not seem to have appeared elsewhere. We also discuss the relationship between ordinary homology and cohomology when the group is compact Lie, in which case the two theories are not represented by the same spectrum. The remainder of the work discusses the extension to grading on representations of fundamental groupoids, concentrating on those aspects that are not simple generalizations of the RO(G)-graded case. These theories can be viewed as defined on parametrized spaces, and then the representing objects are parametrized spectra; we use heavily foundational work of May and Sigurdsson on parametrized spectra. We end with a discussion of Poincare duality for arbitrary smooth equivariant manifolds.
Weights in the cohomology of toric varieties  [PDF]
Andrzej Weber
Mathematics , 2003,
Abstract: We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
Koszul Duality for modules over Lie algebra  [PDF]
Tomasz Maszczyk,Andrzej Weber
Mathematics , 2001,
Abstract: Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.
Localization algebras and deformations of Koszul algebras  [PDF]
Tom Braden,Anthony Licata,Christopher Phan,Nicholas Proudfoot,Ben Webster
Mathematics , 2009,
Abstract: We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_n$ is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category $\mathcal{O}$" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.
String topology prospectra and Hochschild cohomology  [PDF]
Kate Gruher,Craig Westerland
Mathematics , 2007,
Abstract: We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group $G$, we show that the string topology prospectrum $LBG^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify $LBG^{-ad}$ with the homotopy orbit spectrum $(DG)_{hG}$, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of $C^*(BG)$ and $C_*(G)$, respectively. These, in turn, are isomorphic via Koszul duality.
On Koszul duality for Kac-Moody groups  [PDF]
Roman Bezrukavnikov,Zhiwei Yun
Mathematics , 2011,
Abstract: For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in the work of Beilinson, Ginzburg and Soergel .
Equivariant cohomology, syzygies and orbit structure  [PDF]
Christopher Allday,Matthias Franz,Volker Puppe
Mathematics , 2011, DOI: 10.1090/S0002-9947-2014-06165-5
Abstract: Let X be a "nice" space with an action of a torus T. We consider the Atiyah-Bredon sequence of equivariant cohomology modules arising from the filtration of X by orbit dimension. We show that a front piece of this sequence is exact if and only if the H^*(BT)-module H_T^*(X) is a certain syzygy. Moreover, we express the cohomology of that sequence as an Ext module involving a suitably defined equivariant homology of X. One consequence is that the GKM method for computing equivariant cohomology applies to a Poincare duality space if and only if the equivariant Poincare pairing is perfect.
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