From the exact couple formulation of spectral sequences, we introduce a special case of the leray serre spectral sequence and use it to compute h cpn. Abc is the constant functor with value z, as is clear from the isomorphism homcaz, af tm lim af. Equivariant cohomology, localisation and moment graphs university of melbourne june 6th 2011. S1 and we know the cohomology of s3, we can recover the cohomology of u2. The purpose of this thesis is to describe in detail duggers 6 construction of the spectral sequence, which computes krtheory of an equivariant space from its equivariant cohomology. We can apply the serre spectral sequence since we have a simple bration. Equivariant homology and cohomology see also 19l47 keywords spectral sequence algebraic topology local coefficient equivariant homology and cohomology. Okay, now that we know how to calculate with a spectral sequence, where does it come from.
Our approach bypasses the previously mentioned di culties surrounding the homotopy xed point spectral sequence by using the c4 equivariant spectra bp. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. The borel construction of the equivariant cohomology h g x goes as follows. Hodgkin, the equivariant kunneth theorem in ktheory, springer lecture notes. On the spectrum of the equivariant cohomology ring 5 iv the following sequence of h. In this paper, a version of the serre spectral sequence of a fibration is developed for ro g graded equivariant cohomology of g spaces for finite groups g. The spectrum ehc12 4 is one of the building blocks of the k4local sphere lk4s, in a sense that we will make precise in section 1. It expresses, in the language of homological algebra. This shows that these two properties determine the equivariant cohomology up to canonical. However, there has been a notable lack of computations done. This course covers cohomology, poincare duality, homotopy groups, the serre spectral sequence, and the basics of stable homotopy. It is also possible sometimes to run the serre spectral sequence.
As an application we get the serre spectral sequence in the. In bre67, bredon created equivariant homology and cohomology theories of. We then show how to construct new equivariant analogues of these spectral sequences and give a worked example of how to apply them in a computation involving the equivariant serre spectral sequence. We use this new cohomology to construct a serre spectral sequence for equivariant fibrations. Y x be a hurewicz gfibration, that is, a gmap which has the ghomotopy lifting property with respect to. Homology with local coefficients, equivariant cohomology 55n91. We then show how to construct new equivariant analogues of these spectral sequences and give a worked example of how to apply them in a. To see the equality of equivariant cohomologies note that eg c is a contractible free gspace and hence the inclusion g. The slice spectral sequence in equivariant stable homotopy theory is analogous to the slice spectral sequence in motivic homotopy theory see voevodsky6 and was an important tool in the recent solution of the kervaire invariant problem see hill, hopkins and ravenel4. Spectral sequences in algebra, geometry and analysis 507.
Roughly speaking, equivariant formality amounts to the existence of equivariant extension in the equivariant cohomology theory of any element in the ordinary cohomology theory. A spectral sequence for homotopy groups of equivariant mapping spaces. The spectral sequence whose existence is asserted in the above theorem is an example of a. In mathematics, the atiyah hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by michael atiyah and friedrich hirzebruch 1961 in the special case of topological ktheory. Equivariant spectral sequences for local coefficients. Eg0given in item 2 and the identity on x is an isomorphism on the homotopy groups and hence on the cohomology groups. The equivariant serre spectral sequence radboud repository. An essay on spectral sequences university of washington. As a consequence, we show that the equivariant motivic spectral sequence constructed by levineserp\e degenerates rationally which gives a riemannroch theorem for equivariant ktheory. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. A sheaftheoretic approach to the equivariant serre spectral sequence.
The serre spectral sequence is defined for fibrations fxb and relates the. The equivariant serre spectral sequence as an application of. We show that if the basegspacex is equivariantly triangulated and y is paracompact, then spaniers spectral sequence yields an equivariant serre spectral sequence for the main point here is to identify the equivariant singular cohomology groups of x with appropriate simplicial cohomology groups of the orbit space x g. Let ebe a spectral sequence, and suppose that for some r. An algebrogeometric realization of equivariant cohomology. A sheaftheoretic approach to the equivariant serre spectral. The main result of this paper is to extend the spectral sequence. From this thein particular partof thetheoremfollows. Using sheaftheoretic methods, we show that, under suitable assumptions on the gspaces x and y, the equivariant cohomology of y is the limit of a spectral sequence, whose e2term is given by the equivariant cohomology of x with coefficient system depending on the equivariant cohomology of the fibre of f. Bredon 1 introduced what is now called bredon cohomology, with the purpose of developing obstruction theory in the context of spaces equipped with an action of a fixed group g. A gaction on a space xis said to be equivariantly formal if the leray serre spectral sequence for the rational cohomology of the ber bundle x.
The main result of this paper is to extend the spectral sequence of a g. An essay on spectral sequences may 15, 2016 this is primarily an essay on the serre spectral sequence for the homologycohomology of a bration. Most spectral sequences we will encounter will be biregular. Equivariant cohomology, localisation and moment graphs university of melbourne june 6th 2011 monday problem sheet 1. Power operations and the kunneth spectral sequence. Introduction to spectral sequences ucb mathematics. Fibrations and the leray serre spectral sequence 4 3. For a cw complex and a generalized cohomology theory. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. The construction uses the idea of equivariant ktheory of automorphisms to produce a tower for equivariant ktheory, where the successive layers are weak. Equivariant cohomology, localisation and moment graphs. Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences. Cohomology eigenvalues of equivariant mappings springerlink.
In the second chapter, we construct the serre spectral sequence of a. Pdf equivariant cofibrations and nilpotency robert h. Topology and its applications elsevier topology and its applications 90 1998 1119 the equivariant serre spectral sequence as an application of a spectral sequence of spanier hannu honkasalo1 department of mathematics, p. Applications of group cohomology university of texas at austin. For x smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary. Dorette pronk, dalhousie university bredon cohomology with local coe. A construction of equivariant stable homotopy theory in terms of spectral mackey functors is due to. The main result of this paper is to extend the spectral sequence of a. Equivariant spectral sequences for local coefficients arxiv. Asin5, we let d d g bethecategorywhose objectsarethemorphisms u. There is a natural action of g on g by conjugation g.
Let be a real bundle, in the sense of atiyah, over a space x. This was generalized to arbitrary topological groups by illman 2. For spaces with a group action, we introduce bredon cohomology with local or twisted coefficients and show that it is invariant under weak equivariant homotopy equivalence. Ams proceedings of the american mathematical society. We show that if the basegspacex is equivariantly triangulated and y is paracompact, then spaniers spectral sequence yields an equivariant serre spectral sequence for the main point here is to identify the equivariant singular cohomology groups of x with appropriate simplicial cohomology groups of. For this we compute the leray serre spectral sequences with. A users guide to spectral sequences second edition john mccleary vassar college cambridge. Contents 1 preface 2 2 the serre spectral sequence 4 3 spectral sequences in general 22 4 the spectral sequence of a ltered complex 28. In mathematics, in the field of homological algebra, the grothendieck spectral sequence, introduced by alexander grothendieck in his tohoku paper, is a spectral sequence that computes the derived functors of the composition of two functors.
Box 4, fin00014 university of helsinki, helsinki, finland received 3 april 1996. On the spectrum of the equivariant cohomology ring mark goresky and robert macpherson abstract. New york journal of mathematics university at albany, suny. In mathematics, the serre spectral sequence sometimes lerayserre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology. September 1981 equivariant cofibrations and nilpotency by robert h. The equivariant serre spectral sequence as an application.
Pdf hosted at the radboud repository of the radboud university. Describe classifying spaces for the following groups. On a spectral sequence for equivariant ktheory request pdf. Show that the diagonal gaction on e xis free for any gspace x. This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. Cartan model and spectral sequence let g be a compact lie group acting on a smooth compact manifold m di. 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. Svensson, the equivariant serre spectral sequence, \em proceedings of the ams 118 1993, pp. We cover essentially only that part of the theory needed in algebraic geometry. For details on the plain serre spectral sequence see at atiyahhirzebruch spectral sequence and take e h r e h r to be ordinary cohomology. Bert guillou, peter may, models of g gspectra as presheaves of spectra, arxiv. Dorette pronk, dalhousie university bredon cohomology.
Equivariant cohomology of finite group actions and. Apply the leray serre spectral sequence to the bration c 1nf0g. Also, perhaps partially because of a lack of this spectral sequence, the theory of equivariant characteristic classes has not yet been developed. Y x be a hurewicz gfibration, that is, a gmap which has the ghomotopy lifting property with respect to all gspaces. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians.
A sheaftheoretic approach to the equivariant serre. It follows from 1 and 2 that the entry of every subsequent page of the spectral sequence is also zero. If 77is finite then it is not hard to prove that 77,77, m is finite for any finitely generated m show it first for 77 zp, then use the serre spectral sequence ofoz,77a0 and induction. A primer november 1, 2018 1 the borel construction let gbe a discrete group and let g x. The serre spectral sequence or lerayserre spectral sequence is a spectral sequence for computation of ordinary cohomology ordinary homology of topological spaces in a serrefiber sequence of topological spaces. Equivariant cohomology in the framework of smooth manifolds is the subject of this book which is part of a collection of volumes edited by j. Equivalently, a gspace xis equivariantly formal if and only if the leray serre spectral sequence of the ber bundle x. On the slice spectral sequence university of rochester. X denotes the union of the 0 and 1dimensional orbits of t. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the lyndonhochschild serre spectral sequence.
Spectral sequences via examples the graduate journal of. Spectral sequences fabian hebestreit, achim krause, thomas nikolaus june 27, 2017 abstract these are notes for a course taught at the university of bonn in summer 2017. Classifying spaces and characteristic classes 207 6. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology of the equivariant. Quillens theorems on the spectrum of an equivariant cohomology ring, the atiyahswan conjecture, etc. Conjugation spaces and equivariant chern classes wolfgang pitsch and jer ome scherer abstract. It is a spectral sequence of mackey functors which converges strongly. Recall 16 that the action of t on x is equivariantly formal if the leray serre spectral sequence. X g c eg c x g c since the bre is contractible, the projection is a homotopy equivalence. If an algebraic torus t acts on a complex projective algebraic variety x, then the af. In mathematics, the serre spectral sequence sometimes leray serre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology.