For example, in the serre spectral sequence for homology the first few pages. 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. Reference for specific detail on serre spectral sequence. Quantization of the serre spectral sequence jeanfrancois barraud and octav cornea the present paper is a continuation of our earlier work lagrangian intersections and the serre spectral sequence, ann. Lagrangian intersections and the serre spectral sequence 661 from this perspective, it is clear that it is not so important where the spectral sequence efl,l converges but rather whether it contains many nontrivial di.
These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. Note that if a is a rational space and x,a is a relative cw0complex, then x is rational as well. The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. As a consequence, we will derive some homotopytheoretic applications. The serre spectral sequence is a way to compute the cohomology of y. In the second chapter, we construct the serre spectral sequence of a. Use the serre spectral sequence in order to compute the. We prove the above results by using basic facts about the hochschildserre spectral sequence for relative lie algebra cohomology, and most of the paper consists of. We can then form a knlocal ehh nadams resolution of ehk n in the stable category of ehk nmodules, again in the sense of 8. Also an interpretation of the relative cohomology groups for lie algebras in. Serre spectral sequences are a powerful theoretical and computational tool with numerous applications to algebraic topology. There is a fibration xmb with the mth postnikov stage pmf1 as the fiber, and a mapx xmwhich commutes with projection ontobandonfibers induces the usual mapfi.
Spectral sequences are a generalization of exact sequences, and since their introduction by jean leray, they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. We prove the above results by using basic facts about the hochschildserre spectral sequence for relative lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of hochschild and serre. The fundamental example of a spectral sequence, from which essentially all the other examples arise as special cases, is the spectral sequence of a filtered complex. This follows directly from a relative version of the serre spectral sequence. On the finite generation of relative cohomology for lie superalgebras andrew maurer abstract. Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry.
More about the homotopy groups of spheres use the serre spectral sequence and the hurewicz theorem to show that. This can be proved by a standard application ofthe relative serre spectral sequence. Spanier 10 that this sequence can be derived from the serre spectral sequence. The relative hochschildserre spectral sequence and.
Group cohomology and the hochschildserre spectral sequence. This does not have a ring structure, but it does have a module structure over the previous spectral sequence. Asking for help, clarification, or responding to other answers. This naturally suggests that the rings and he are isomorphic, but this conclusion is, in general, false.
Discuss in detail the multiplicative structure in the spectral sequence pages 543546. Ehs eckmannhiltonstammbach,on the homology theory of central group extensions ii. This leads to a spectral sequence analogous to the one by fukaya, seidel, and smith with a. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. Fibrations and the lerayserre spectral sequence 4 3. Thanks for contributing an answer to mathematics stack exchange.
Serre spectral sequence in cohomology, by constructing the appropriate staircase diagram. 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. A primer on spectral sequences university of chicago. We construct using relatively basic techniques a spectral sequence for exact lagrangians in cotangent bundles similar to the one constructed by fukaya, seidel, and smith. Band relates the homology of f, x, and b, under an added technical hypothesis which is satis. Its main focus is the serre spectral sequence and its applications, but there is also some coverage of the adams spectral sequence and, more briefly, a few other spectral sequences. Serre spectral sequences and applications emory computer. Pdf the relative hochschildserre spectral sequence and. A spectral sequence is constructed to provide conditions for when this relative cohomology ring is cohenmacaulay. 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 spectral sequence is a sequence of double complexes er pq,d r with di. Kathryn hessbellwald supervisedby marc stephan academic year.
Homotopy equivalence of nearby lagrangians and the serre spectral sequence thomas kragh abstract. Rational homotopy theory is the study of rational ho. On the finite generation of relative cohomology for lie. A lyndonhochschildserre spectral sequence 3 arising from functoriality are maps of commutative s0algebras. The homology of this is, by definition, the relative homology.
X,fb, b, which we distinguish from the original spectral sequence by using. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. Thus, in codimension 1, for example, if the fibers are all compact and connected, 2 reduces to the thomgysin sequence of a circle bundle, and it is known cf. In the second chapter, we construct the serre spectral sequence of a fibration. The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the lyndonhochschildserre spectral sequence. Homotopy equivalence of nearby lagrangians and the serre. September reading responses graduate topology seminar. Let ebe a spectral sequence, and suppose that for some r. It will describe a very particular relation between the homology groups. In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations.
Relative lie algebra cohomology revisited mark muzere communicated by maurice auslander abstract. In case you would like to see the older version that was planned as a separate book, this can be found on this page. Most spectral sequences we will encounter will be biregular. The hochschildserre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature. Associated to a fibration there is the serre spectral sequence 10. 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 cover essentially only that part of the theory needed in algebraic geometry. Long exact sequence in h through a range for a bration 27 x2 application ii. 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. The class uin the above theorem is called the thom class. There are two kinds of relative serre spectral sequences. Lagrangian intersections and the serre spectral sequence. Topological library series on knots and everything. A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr.
The spectral sequence whose existence is asserted in the above theorem is an example of a. Letxmbethe mthstage in the moorepostnikovfactorization of the map xb8, p. The exact sequence in homotopy groups, and the leray serre spectral sequence for homology groups 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. One of the main desirable properties of the serre spectral sequence is to account for the action of the fundamental. These are the lecture notes for a minicourse on spectral sequences held at.
The relative hochschildserre spectral sequence and the. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians. As an application, derive the cohomology ring of kz. Also, the construction of the relative hochschildserre spectral sequence in bw uses the identi. In this paper it is shown that relative lie algebra cohomology is related to relative cohomology for restricted lie algebras by a spectral sequence. That spectral sequence was used to prove that exact relative spin lagrangians in simply connected cotangent bundles with vanishing maslov class are homology equivalent to the base a similar result was also obtained by. Introduction to spectral sequences ucb mathematics. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \spectral sequence strike fear into the hearts of many hardened mathematicians.
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