Multiplicative structure of the atiyahhirzebruchlerayserre. Lerayserre spectral sequence for quasifibrations is that hurewicz theorem relates both up to some extent, which may help us to compute homotopy groups. 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. 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. Here we can apply two different but related lerayserre spectral sequences with inter mediate step h. In section 4, we will do all of the homological algebra groundwork and construct the spectral sequence of a ltered chain complex. Introduction to spectral sequences ucb mathematics. A users guide to spectral sequences by john mccleary. The exact sequence in homotopy groups, and the leray serre spectral sequence for homology groups of a. Fibrations and the leray serre spectral sequence 4 3. Is the serre spectral sequence a special case of the leray.
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. Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. It is also possible sometimes to run the serre spectral sequence backwards, if. 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. Some extremely brief notes on the leray spectral sequence. It is constructed from the following filtration of the total space e. Fibered symplectic cohomology and the lerayserre spectral. The maps forming chain complexes on each page are known as. Jardine mathematics department, university of western ontario, london, ontario n6a 5b7, canada communicated by e. 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. Greg friedman, some extremely brief notes on the leray spectral sequence. Eg just as there is o les of a pair xa in homology. 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.
When the base is symplectically aspherical we construct a spectral sequence of lerayserre type converging to the. In fact, the result is basically obvious if you use czech cohomology on the base. Multiplicative structure in the cohomological lerayserre. The leray spectral sequence, the way we proved it in lemma 20. 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. Journal of pure and applied algebra 61 1989 189196 189 northholland the leray spectral sequence j.
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. A question about the lerayserre spectral sequence mathoverflow. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. There he took up algebraic topology, and the result was a spectacular.
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. Spectral sequence was invented to handle the complicated relationship among homology. Lerayserre spectral sequence for quasifibrations 1 homotopy. Dan petersen, leray spectral sequence, november 2010. The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. In the second chapter, we construct the serre spectral sequence of a. We already know how to do this in case of the trivial bre sequence f. 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. E b be a continuous map of topological spaces say of cw complexes. Assumingthat bissimplyconnected,serre proved that there is a spectral sequence with e. Coce yo q kc acy ces sequaoce qssocxcqd toe sequonce a pays, pcxca2 \thqs. Spectral sequences via examples the graduate journal of.
Xby means of he kunneth theorem, namely that there is a split short exact sequence 0. A short exact sequence of chain complexes gives rise to a long exact sequence. Leray spectral sequence converging to the cohomology of a sheaf. An important example of exact couple comes from the long. 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. 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. It has been suggested that the name spectral was given because, like spectres. Ams transactions of the american mathematical society. In section 5, we will translate this to topology and construct the serre spectral sequence. Though the abutment an is the same, the filtration is, in general, different. 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. Spectral sequences and applications laurentiu maxim department of mathematics. One can place this in a wider setting as an example of a coloured.
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. Basic definitions a spectral sequence is a construction that enables us to form from two graded abelian groups or modules over a ring. For a comprehensive introduction to spectral sequences, see 3. Dan petersen, leray spectral sequence, november 2010 greg friedman, some extremely brief notes on the leray spectral sequence. A lerayserre spectral sequence for lagrangian floer theory. 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.
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. 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. It was first invented by the french mathematician jean leray from. Bundles of coloured posets and a leray serre spectral sequence for khovanov homology brent everitt and paul turner abstract. 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. 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. 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. We cover essentially only that part of the theory needed in algebraic geometry. The serre spectral sequence is a way to compute the cohomology of y in terms of the cohomology of xand f. The lerayserre spectral sequence is a spectral sequence with. There are more general data giving rise to spectral sequences. May spectral sequence for calculating the tor or ext groups of an algebra.
This chapter is devoted to the lerayserre spectral sequence for the homology and cohomology of a fibration. A di erential bigraded module over a ring r, is a collec. The spectral sequence whose existence is asserted in the above theorem is an example of a. There are essentially two situations in which spectral sequences arise. 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. We then show how this theory finds an application in khovanov homology by producing a new spectral sequence converging to the. The lerayserre spectral sequence in morse homology on. The lerayserre spectral sequence for the cohomology and the homology of a fibration is introduced and the e2 term is evaluated.
Then it is possible to construct the lerayserre spectral sequence in purely morsehomological terms. Some extremely brief notes on the leray spectral sequence intro. Assumingthat bissimplyconnected, serre proved that there is a spectral sequence with e. 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. In a sense there is really only one spectral sequence, just as there is only one concept of a long exact sequence although each object may originate in a variety of settings, but there are many di. For example, in the serre spectral sequence for homology the. An introduction to spectral sequences matt booth december 4, 2016. 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. The cohomology spectral sequence multiplicative structure. A leray spectral sequence for noncommutative differential fibrations article pdf available in international journal of geometric methods in modern physics 105.
Lyndonhochschildserre spectral sequence in group cohomology. Fibrations and the lerayserre spectral sequence 4 3. Miller spectral sequence converging to the mod p stable homology of a space. 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. This is part one of authors phd thesis in the form of a shortened paper and with a more refined perturbation system. Such a spectral sequence may have more utility in a theoretical setting i. Leray gave a hint of how he came to the notion of a spectral sequence in a later paper 47, p. Multiplicative structure in the cohomological lerayserre spectral sequence please elucidate a proof. The grothendieck spectral sequence minicourse on spectral. Multiplicative structure of the atiyahhirzebruchleray. Bwith trivial monodromy on the homology of the bers, e2 p.
Notes on spectral sequence northeastern university. 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. As a consequence, we will derive some homotopytheoretic applications. Spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. Textbook accounts with an eye specifically towards etale cohomology. Bundles of coloured posets and a lerayserre spectral sequence for khovanov homology brent everitt and paul turner abstract. They were introduced by leray in the 1940s at the same time as he introduced sheaves. 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.
They have a reputation for being abstruse and difcult. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. 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 spectral sequence is a computational tool they are more compiex analogues of long exact sequences.