These are two simple conditions that force the leray spectral sequence to converge. Let hbe a cohomology theory and xbe a nite cw complex. Generalizing the lerayhirsch theorem the proof of the lerayhirsch theorem as given in 9, p. This entry is a detailed introduction to stable homotopy theory, hence to the stable homotopy category and to its key computational tool, the adams spectral sequence. Altogether, these additional topics amount to nearly half the book, and they are included here both to make the book more comprehensive and to give the reader who takes the time to delve into them a more substantial sample of the true richness and beauty of the subject. The main idea is to make a spectral decomposition of am, e, i 1, 2 in terms of the behaviour of the eigenvalues of a as e 0. The point of lerayhirsch is that it is sometimes free. 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 order to apply the leray hirsch theorem, we must verify point 3. Homotopy theory, examples 3 oscar randalwilliams lent 2015 1. So in a sense, the serre spectral sequence can be seen as measuring the obstruction to finding cohomology classes that restrict to generators of the fiber. In these cases n2n is the dimension of the manifold. Peter kronheimer taught a course math 231br on algebraic topology and algebraic k theory at harvard in spring 2016. Pdf degeneration of the leray spectral sequence for certain. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. In order to apply the lerayhirsch theorem, we must verify point 3. One is the splitting principle, the other one is the lerayhirsch theorem. Some extremely brief notes on the leray spectral sequence intro. Lectures on perverse sheaves and decomposition theorem.
Projective toric generators in the unitary cobordism ring. The hypercohomology groups of a boundedbelow complex generalize the ordinary cohomology groups on an object. Pdf hosted at the radboud repository of the radboud. Free algebraic topology books download ebooks online. Chain complexes, homology, and cohomology, homological algebra, products, fiber bundles, homology with local coefficient, fibrations, cofibrations and homotopy groups, obstruction theory and eilenbergmaclane spaces, bordism, spectra, and generalized homology and spectral sequences. The classical theorems of leray and hirsch and of borel and serre are. Use the lerayserre spectral sequence for the homotopy bration cp1. We specialize to the case of free loop spaces where the embedding is the diagonal embedding. He studied at ecole normale superieure from 1926 to 1929. We find some spectral sequences to compute the homology of x in terms of the homology of y and that of the fibers. Lerayserre spectral sequence for quasifibrations 1 homotopy. We also prove a twisted lerayhirsch theorem, which we will make repeated use of in later sections. Explicit computations for the intersection of two complex quadrics are presented.
The lerayhirsch theorem states sufficient fiberwise condition for the ordinary cohomology of the total space of a fiber bundle with coefficients in a commutative ring to be free module over the cohomology ring of the base space. In fact, a similar sequence exists for a bration f. The leray hirsch theorem states sufficient fiberwise condition for the ordinary cohomology of the total space of a fiber bundle with coefficients in a commutative ring to be free module over the cohomology ring of the base space. In mathematics, the lerayhirsch theorem is a basic result on the algebraic topology of fiber bundles. Cohomology of symmetric spaces of classical type 146 chapter 4. As an example of how this could be used, we point out that for any. Use the cohomological lerayserre spectral sequence and the derivation property to compute d 3 explicitly, using the cohomology ring of cp1. S2 s2 be the pullback of this bre bundle along a map s2 s2. Consider the lerayserre spectral sequence for the path fibration. Spectral sequences and applications department of mathematics. The proof is based on the borel cohomology adams spectral sequence and the calculations with the steen.
We know that if we have a short exact sequence of chain complexes 0 c. In the same year, he and juliusz schauder discovered a topological invariant, now called the lerayschauder. In 1934 leray published an important paper that founded the study of weak solutions of the navierstokes equations. The connection of brownkervaire invariants to the adams spectral sequence examined in 22 is only included for the sake of completeness. It can be thought of as a mild generalization of the kunneth formula, which computes the cohomology of a product space as a tensor product of the. Pdf degeneration of the leray spectral sequence for. Since zp is the psylow subgroup of p, and since g is the normalizer of that in p, the cohomology of g is the cohomology of p, plocally. May 24, 2010 the e2 page of the serre spectral sequence is what one would obtain by leray hirsch. The e2 page of the serre spectral sequence is what one would obtain by lerayhirsch. If uu i is an open cover of y, form the cech complex with respect to cover f. Singular homology of fiber spaces topological library. Splitting principle and the projective bundle formula there are two leftovers from the proof of the existence and uniqueness of stiefelwhitney classes. Cartanleray and lyndonhochschildserre spectral sequences, are.
The proof, which is premised on a few basic properties of the serre spectral sequence, is deferred to a later point in the course, at which we will discuss this tool in some detail. Introduction to the serre spectral sequence, with a number of applications, mostly fairly standard, the adams spectral sequence, eilenbergmoore spectral sequences. For cohomology with rational coe cients, the same proof applies to a slightly more general situation. Lecture notes in algebraic topology pdf 392p download book. We will start with some general concepts and results about fiber bundles and vector bundles, then we will move to topics like topological ktheory, stiefelwhitney. Characteristic classes and the relations among them 5 6. It is a very special case of the leray spectral sequence. The main idea is to make a spectral decomposition of am, e, i 1, 2 in. This is the third authors lecture in stringmath 2015 at sanya, which reports on our recent works in quantum cohomology. He was born in chantenaysurloire today part of nantes. Cohomology of fibrations the serre spectral sequence 105 3. Lecture notes in algebraic topology pdf 392p this note covers the following topics.
Lecture notes on homotopy theory and applications math. As a consequence, we will derive some homotopytheoretic applications. By a spectral sequence in an abelian category a, we mean a collection of bi. The basic examples of manifolds with corners of codimension kfor 0 k nare open subsets of 0. This result agrees with what was previously known about the cohomology of m4. Qis the same as the cohomology of a projective bundle. Each lecture gets its own chapter, and appears in the table of contents with the date. As an intermediate step, we obtain a stacky version of the very useful leray hirsch theorem. A nontrivial ghost kernel for the equivariant stable cohomotopy of projective spaces markus szymik abstract it is shown that the ghost kernel for certain equivariant stable cohomotopy groups of projective spaces is nontrivial.
Free algebraic topology books download ebooks online textbooks. Some extremely brief notes on the leray spectral sequence. Homotopy theory, examples 3 university of cambridge. It is named after jean leray and guy hirsch, who independently proved it in the late 1940s. As a motivating example, consider the long exact homology sequence. Topological library series on knots and everything. As an intermediate step, we obtain a stacky version of the very useful lerayhirsch theorem. Spectral sequence of a differential group with increasing filtration.
S3 to calculate h s3h3i and use the result to show that. The lerayhirsch theorem and the action of the weyl group 126 5. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. Thakur assume that all endomorphisms of the holomorphic vector bundles e1 and e2 are scalar multiplications. In the accompanying seminar we consider applications to cobordism theory and complex oriented cohomology such as to converge in. The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. Let e be a vector bundle of weight 1 and rank n c on x.
We will give an introduction to vector bundles, characteristic classes, and spectral sequences. Pdf the heredity and bimeromorphic invariance of the. Ag 25 jan 2002 degeneration of the leray spectral sequence for certain geometric quotients c. Lerayhirsch theorem for dolbeault cohomology mathoverflow. This is the third course in the first year graduate algebraic topology sequence. For a proof of the existence of a spectral sequence under the conditions alluded to above, see grothendieck spectral sequence. In section 8, as an example, we consider the case when the embedding is the inclusion of complex projective spaces. Lerayhirsch theorem and the contraction of curves on surfaces. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. It can be thought of as a mild generalization of the kunneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors.
Algebraic topology lectures by haynes miller notes based on livetexed record made by sanath devalapurkar images created by john ni march 4, 2018 i. So, if the leray spectral sequence degenerates at e2, knowing the cohomology of the source x is equivalent to knowing that of the target y. This particular statement is a corollary of the general lerayhirsch theorem about the degeneration of the serre spectral sequence and the vanishing of cn. Recently active spectralsequences questions page 3. I am asking this because i know that lerayhirsch theorem is a very particular case of leray spectral sequence which is valid for sheaf cohomology in general.
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