Such a spectral sequence may have more utility in a theoretical setting i. Introduction to spectral sequences ucb mathematics. From the exact couple formulation of spectral sequences, we introduce a special case of the lerayserre spectral sequence and use it to compute h cpn. A lerayserre spectral sequence for lagrangian floer theory 3 and a lagrangian lb is called rational if the set of values z d u b. First, sheaf cohomology is a pain to compute, but if there is no fundamental group then for fiber bundles the leray spectral sequence is really just using normal old. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. Textbook accounts with an eye specifically towards etale cohomology. Journal of pure and applied algebra 61 1989 189196 189 northholland the leray spectral sequence j. Bwith trivial monodromy on the homology of the bers, e2 p. Bundles of coloured posets and a leray serre spectral sequence for khovanov homology brent everitt and paul turner abstract.
Multiplicative structure in the cohomological lerayserre spectral sequence please elucidate a proof. We then derive a spectral sequence in the vein of the classical leray serre sequence and show that it converges to the lagrangian floer cohomology. Spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. May spectral sequence for calculating the tor or ext groups of an algebra. For a comprehensive introduction to spectral sequences, see 3. When the base is symplectically aspherical we construct a spectral sequence of lerayserre type converging to the. A short exact sequence of chain complexes gives rise to a long exact sequence. One can place this in a wider setting as an example of a coloured. We cover essentially only that part of the theory needed in algebraic geometry. Dan petersen, leray spectral sequence, november 2010 greg friedman, some extremely brief notes on the leray spectral sequence. The lerayserre spectral sequence is a spectral sequence with. The heart of the book is an exposition of the classical examples from homotopy theory, with chapters on the lerayserre spectral sequence, the eilenbergmoore spectral sequence, the adams spectral sequence, and, in this new edition, the bockstein spectral sequence. The cohomology spectral sequence multiplicative structure.
Lerayserre spectral sequence for quasifibrations is that hurewicz theorem relates both up to some extent, which may help us to compute homotopy groups. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a lerayserre type spectral sequence. Russian mathematical surveys lectures at the mathematical. It was first invented by the french mathematician jean leray from. Multiplicative structure in the cohomological lerayserre. In section 5, we will translate this to topology and construct the serre spectral sequence. It expresses, in the language of homological algebra, the singular cohomology of the total space x of a serre fibration in terms of the cohomology of the base space b and the fiber f. There are more general data giving rise to spectral sequences. An important example of exact couple comes from the long. 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. Ams transactions of the american mathematical society. Some extremely brief notes on the leray spectral sequence intro. There are essentially two situations in which spectral sequences arise.
The exact sequence in homotopy groups, and the leray serre spectral sequence for homology groups of a. Spectral sequences are introduced in the general context of filtered chain and cochain complexes. A spectral sequence is a computational tool they are more compiex analogues of long exact sequences. Spectral sequences and applications laurentiu maxim department of mathematics. This chapter is devoted to the lerayserre spectral sequence for the homology and cohomology of a fibration. First, sheaf cohomology is a pain to compute, but if there is no fundamental group then for fiber bundles the leray spectral sequence is really just using normal oldfashioned untwisted cohomology. Multiplicative structure of the atiyahhirzebruchleray. In section 4, we will do all of the homological algebra groundwork and construct the spectral sequence of a ltered chain complex. In the second chapter, we construct the serre spectral sequence of a. Basic definitions a spectral sequence is a construction that enables us to form from two graded abelian groups or modules over a ring and f a third group or module.
A leray spectral sequence for noncommutative differential fibrations article pdf available in international journal of geometric methods in modern physics 105. The grothendieck spectral sequence minicourse on spectral. A lerayserre spectral sequence for lagrangian floer. Eg just as there is o les of a pair xa in homology. Then it is possible to construct the lerayserre spectral sequence in purely morsehomological terms. The decorated hypercube found in the construction of khovanov homology for links is an example of a boolean lattice equipped with a presheaf of modules. In fact, the result is basically obvious if you use czech cohomology on the base. The maps forming chain complexes on each page are known as. It is constructed from the following filtration of the total space e. There he took up algebraic topology, and the result was a spectacular. Assumingthat bissimplyconnected,serre proved that there is a spectral sequence with e.
We then show how this theory finds an application in khovanov homology by producing a new spectral sequence converging to the. Fibrations and the lerayserre spectral sequence 4 3. Dan petersen, leray spectral sequence, november 2010. Spectral sequences via examples the graduate journal of. Lyndonhochschildserre spectral sequence in group cohomology.
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. A di erential bigraded module over a ring r, is a collec. They were introduced by leray in the 1940s at the same time as he introduced sheaves. An introduction to spectral sequences matt booth december 4, 2016. Xby means of he kunneth theorem, namely that there is a split short exact sequence 0. Multiplicative structure of the atiyahhirzebruchlerayserre. E b be a continuous map of topological spaces say of cw complexes. Is the serre spectral sequence a special case of the leray. The serre spectral sequence or leray serre spectral sequence is a spectral sequence for computation of ordinary cohomology ordinary homology of topological spaces in a serre fiber sequence of topological spaces. Friedlander received 26 may 1988 revised 7 november 1988 the purpose of this note is to construct a leraytype spectral sequence for homotopy classes of maps of.
The origins of sheaf theory, sheaf cohomology, and spectral sequences haynes miller february 23, 2000 jean leray november 7, 1906november 10, 1998 was con ned to an o cers prison camp \o ag in austria for the whole of world war ii. Bundles of coloured posets and a lerayserre spectral sequence for khovanov homology brent everitt and paul turner abstract. Leray gave a hint of how he came to the notion of a spectral sequence in a later paper 47, p. Ernie presented a version of this sequence on wednesday the version i give will look a little more general, as i will consider more general maps and will allow coe. Miller spectral sequence converging to the mod p stable homology of a space. It is also possible sometimes to run the serre spectral sequence backwards, if. Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. The heart of the book is an exposition of the classical examples from homotopy theory, with chapters on the leray serre spectral sequence, the eilenbergmoore spectral sequence, the adams spectral sequence, and, in this new edition, the bockstein spectral sequence. A users guide to spectral sequences by john mccleary. We show tranversality and compactness for the moduli of pearly morse trajectories on a lagrangian in a compact symplectic fiber bundle with rational base and small monotone fibers. Fibrations and the leray serre spectral sequence 4 3. A short exact sequence of chain complexes gives rise to a long exact sequence in homology, which is a fundamental tool for computing homology in a number of situations.
The lerayserre spectral sequence for the cohomology and the homology of a fibration is introduced and the e2 term is evaluated. The serre spectral sequence is a way to compute the cohomology of y in terms of the cohomology of xand f. The leray spectral sequence, the way we proved it in lemma 20. What i would like to describe today is lerayserre spectral sequence. Notes on spectral sequence northeastern university. A leray serre spectral sequence for lagrangian floer theory 3 and a lagrangian lb is called rational if the set of values z d u b. Jardine mathematics department, university of western ontario, london, ontario n6a 5b7, canada communicated by e. Some extremely brief notes on the leray spectral sequence. Greg friedman, some extremely brief notes on the leray spectral sequence. As a consequence, we will derive some homotopytheoretic applications. The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. Coce yo q kc acy ces sequaoce qssocxcqd toe sequonce a pays, pcxca2 \thqs. Basic definitions a spectral sequence is a construction that enables us to form from two graded abelian groups or modules over a ring.
Though the abutment an is the same, the filtration is, in general, different. For more nice explanations of spectral sequences, see 1 and 2. Assumingthat bissimplyconnected, serre proved that there is a spectral sequence with e. It has been suggested that the name spectral was given because, like spectres. They have a reputation for being abstruse and difcult. Serre, who needed a spectral sequence in homology that applied to path space fibrations, whose total spaces are almost never locally compact, thus was unable to use the original leray spectral sequence and so derived a related spectral sequence whose cohomological variant agrees, for a compact fiber bundle on a wellbehaved space with the. For example, in the serre spectral sequence for homology the. A lerayserre spectral sequence for lagrangian floer theory. In this work, we build a spectral sequence in motivic homotopy that is analogous to both the serre spectral sequence in algebraic topology and the leray spectral sequence in algebraic geometry. The lerayserre spectral sequence in morse homology on. Here we can apply two different but related lerayserre spectral sequences with inter mediate step h. The vast majority of spectral sequences are e 2page spectral sequences, which is to say you begin with the data of the entries on thee 2page. Constructing generalised leray spectral sequences imma g alvez preliminary report, with f neumann and a tonks motivation construction of the first spectral sequence this project arose as a spino of an earlier one with the same collaborators to extend to some categories of stacks several classical results from geometry and topology. The spectral sequence whose existence is asserted in the above theorem is an example of a.
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. Fibered symplectic cohomology and the lerayserre spectral. Dec 22, 2018 in this work, we build a spectral sequence in motivic homotopy that is analogous to both the serre spectral sequence in algebraic topology and the leray spectral sequence in algebraic geometry. We already know how to do this in case of the trivial bre sequence f.
Aug 25, 2011 a leray spectral sequence for noncommutative differential fibrations article pdf available in international journal of geometric methods in modern physics 105 august 2011 with 76 reads. A question about the lerayserre spectral sequence mathoverflow. Lerayserre spectral sequence for quasifibrations 1 homotopy. It is impossible to describe everything about spectral sequences in the duration of a single course, so we will focus on a special and important example. In section 6, we will discuss the product structure of the cohomological serre spectral sequence. Leray spectral sequence converging to the cohomology of a sheaf. This is part one of authors phd thesis in the form of a shortened paper and with a more refined perturbation system.