In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. On the spectral sequence for the equivariant cohomology of a. The spectral sequence converges to the poisson cohomology of the manifold and has the e 2term equal to the tensor product of the cohomology of the lie algebra and the equivariant poisson cohomology of the manifold. Equivariant cohomology in algebraic geometry 3 proposition 2. The main spectral sequence let, be any equivariant cohomology theory and let k be a g complex of dimension n 19. Vector fields, torus actions and equivariant cohomology jim carrell and kiumars kaveh and. We give interesting cohomology classes in the new equivariant cohomology theory.
By constructing two spectral sequences converging to ro ggraded cohomology groups and cech hypercohomology groups, respectively, we show that, for any equivariant good cover u of x and n. On a spectral sequence for equivariant ktheory arxiv. Discussion of the ahss in bredon equivariant stable homotopy theoryequivariant cohomology includes. Equivariant cohomology, koszul duality, and the localization. Any action of a compact lie group k on a compact manifold m with hoddm 0 is equivariantly formal. This spectral sequence is a poisson analog of the leray spectral sequence of a fibration. On a spectral sequence for equivariant ktheory request pdf. As an application, we show that the cohomology furnishes obstructions to equivariant. Pdf the equivariant spectral sequence and cohomology with. This question is a followup to my previous question. The main spectral sequence let, be any equivariant cohomology theory and let k be a g complex of dimension n spectral sequence for equivariant cohomology. We derive a spectral sequence, in the abelian case, which is analogous to the wellknown spectral sequence for the cartan model. For orbifolds there is a generalization of k ktheory which is closely related to the bredon cohomology rather than usual equivariant cohomology.
The equivariant serre spectral sequence 267 the category agx is in some sense the equivariant analogue of the category at associated to a topological space t. Given a polytope in rn, perhaps decorated with additional information, is there a hamiltonian tspace for which this is the image of a moment. For a cw complex and a generalized cohomology theory, it relates the generalized cohomology groups with ordinary cohomology groups with coefficients in the. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. An introduction to equivariant cohomology and the equivariant. We then present the variety x as a homogeneous variety, and so obtain the cohomology. There are various equivariant cohomology theories that may be define. The equivariant spectral sequence and cohomology with local coefficients. If bg is simply connected as is the case for all com. This spectral sequence generalizes the spectral sequence from motivic cohomology to ktheory constructed by blochlichtenbaum and friedlandersuslin. Request pdf on a spectral sequence for equivariant ktheory we apply the machinery developed by the firstnamed author to the ktheory of coherent gsheaves on a finite type gscheme x over a.
When the chain complex is the first page of the equivariant spectral sequence associated to a regular abelian cover of a finitetype cwcomplex, we relate those jump loci to the resonance varieties associated to the cohomology ring of the space. I give a construction which makes no such assumption on x. The cohomology ring of the group is the special case when x is a point. We assume that our cohomology theory h g has suspension isomorphisms for a suit able set of representations, i. Equivariant poisson cohomology and a spectral sequence. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. On the spectrum of the equivariant cohomology ring 5 iv the following sequence of h. A primer november 1, 2018 1 the borel construction let gbe a discrete group and let g x. Applying the gap method to the spectral sequence in item 3 of proposition 3. Hirzebruch spectral sequence from the gequivariant higher chow groups to the higher ktheory of coherent gsheaves on x. Many generalizations of this spectral sequence have been developed for studying cohomology theories in the equivariant context and we will pay speci c attention to the spectral sequence developed by segal in 26. The purpose of this section is to recall the spectral sequence associated to a holomorphic vector eld cl1. This has been done in the past i, when x is a gwcomplex, by considering the filtration ofx by its skeletons.
Conclude that there is a spectral sequence e 2 h x h t pt h x. Hirzebruch spectral sequence from the g equivariant higher chow groups to the higher ktheory of coherent gsheaves on x. Apart from this my purpose is to obtain for a generalized cohomology theory k a spectral sequence connecting ax with the ordinary cohomology of x. Equivariant cohomology of moduli spaces of genus three curves.
This exercise explores the relationship between cohomology and equivariant cohomology and requires some knowledge of spectral sequences. Mike shulman, spectral sequences 20 equivariant version. This cohomology ring is the cohomology of the space xs1 which is the total space of the borel fibering 1,3 the spectral sequence en 1 r oo, of this fibering is such that eoo is the sum of. Let x denote a connected compact kaehler manifold of dimension. A sheaftheoretic approach to the equivariant serre spectral. This cohomology ring is the cohomology of the space xs1 which is the total space of the borel fibering 1,3 the spectral sequence en 1. In mathematics, the atiyahhirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by michael atiyah and friedrich hirzebruch in the special case of topological ktheory. Let eg be a cw complex on which gacts from the left freely and discretely. As an application, we show that vanishing resonance implies a certain. Equivariant cohomology of grassmannians ii william fulton notes by dave anderson 1 as before, let x grk,n, let. Equivariant cohomology, koszul duality, and the localization theorem 27. The moment map and equivariant cohomology sciencedirect.
The tool of equivariant cohomology on a manifold the hirzebruch surface of complex dimension 2 described in example 14 in subsection 7. Rotated version of the atiyahhirzebruch spectral sequence in that question, i discussed two different spectral sequences for. If bg is simply connected as is the case for all compact connected lie groups, the e 2term of the spectral sequence reads 5 ep,q 2 h pbg. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology. Honkasalo, a sheaftheoretic approach to the equivariant serre spectral sequence, j. Discussion of the ahss in bredon equivariant stable. Nov 19, 2018 we study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. Pdf the equivariant spectral sequence and cohomology. Usingsheaftheoreticmethods,weshowthat,undersuitable assumptions onthe gspaces x and y, the equivariant cohomology of y isthelimitofaspectralsequence,whose e2termisgivenbythe. On the spectral sequence for the equivariant cohomology of. This means that for every point x2egthere is a neighborhood. If a group g operates on a topological space x, then one can define equivariant homology and cohomology groups, which can be thought of heuristically as a mixture of hg and hx. Hirzebruch spectral sequence which was originally developed in 4 in order to study ktheory.
Applications of equivariant cohomology 639 let us compute the coef. Equivariant cohomology and the cartan model university of toronto. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. We study the homology jump loci of a chain complex over an affine \\kalgebra.
This equivariant theory provides a powerful tool for extracting homological information about g from the action of g on x. Cequivariant cohomology of aand g c is a big group so bg c might be quite complicated. The equivariant spectral sequence and cohomology with local. Pdf jump loci in the equivariant spectral sequence. We will use this technique to study the spectral sequence that computes equivariant cohomology. A gequivariant differential form is a smooth polynomial a. The use of gspaces and equivariant cohomology was inspired by.
We give interesting cohomology classes in the new equivariant cohomology theory that have no classical analogue. Equivariant cohomology in symplectic geometry rebecca goldin cornell unviersity topology festival may 3, 2008 rebecca goldin gmu equivariant cohomology 1 37. Equivariant spectral sequences for local coefficients. We now describe a particular of the localization process. We introduce and study a new spectral sequence associated with a poisson group action on a poisson manifold and an equivariant momentum mapping. The construction of motivic cohomology as well as the motivic spectral sequence had soon become central problems in the theory of motives and algebraic ktheory during the 1990s and 2000s. Tokyo 4 1997, 5365 for orbifolds there is a generalization of k k theory which is closely related to the bredon cohomology rather than usual equivariant cohomology. A discussion of the e eahss as the spectral sequence of a tower induced by forming mapping spectra x. Equivariant cohomology suppose a topological group g acts continuously on a topological space m. The rst motivic spectral sequence was constructed by blochlichtenbaum for spectrum of.
Quillens theorems on the spectrum of an equivariant cohomology ring, the atiyahswan conjecture, etc. A sheaftheoretic approach to the equivariant serre spectral sequence byhannuhonkasalo abstract. An indispensable tool for equivariant cohomology theory is spectral sequence. This is a survey on the equivariant cohomology of lie group actions on manifolds. The latter is defined as the equivariant cohomology of the multivector fields made into a gdifferential complex by means of the. It follows that, associated to an equivariant good cover of a gmanifold x, there is a bounded spectral sequence converging to cech hypercoho mology whose e 1 page is isomorphic to the e 1 page of a segal spectral sequence which converges to the bredon roggraded equivariant cohomology. It is in this way, for example, that quillen proved his theorem about the. If a group g operates on a topological space x, then one can define equivariant homology and cohomology groups, which can be thought of heuristically as a. Definitions and basic properties william fulton notes by dave anderson 1 for a lie group g, we are looking for a right principal gbundle eg bg, with eg contractible. The equivariant cohomology of isotropyactions on symmetric. The borel construction of the equivariant cohomology h g x goes as follows. Vector fields, torus actions and equivariant cohomology. We study the homology jump loci of a chain complex over an affine \kalgebra. Such a bundle is universal in the topological setting.
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