We shall not be able to avoid using spectral sequences see pp 307309 of my book on etale cohomology for a brief summary of spectral sequences and chapter 5 of weibels book for a complete treatment. The spectral sequence whose existence is asserted in the above theorem is an example of a. In fact, the result is basically obvious if you use czech cohomology on the base. 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. In particular, let us fix such a multiplicative cohomology theory h and let us consider a smooth manifold x of dimension n and a compact submanifold y of dimension p, satisfying suitable. S3 to calculate h s3h3i and use the result to show that.
In particular, the groups of the starting step of the spectral sequence ep,q. No homotopy type theory is required, although anyone whos read through chapter 8 of the hott book ought to be able to translate into hott pretty easily. This turns out to be much more powerful than the serre spectral sequence in homology, because the cup product structure on the e2page of the spectral sequence extends to a product structure on every page of the spectral sequence. 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. The serre spectral sequence is a way to compute the cohomology of y in terms of the cohomology of xand f. The di erentials dsatisfy a leibniz rule with respect to these products, and. The thomgysin sequence is a type of long exact sequence in cohomology induced by a spherical fibration and expressing the cohomology groups of the total space in terms of those of the base plus correction.
Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. There are two eilenbergmoore spectral sequences that we. What i would like to describe today is lerayserre spectral sequence. The first part, in which i attempted to motivate the notion of spectral sequence, and constructed the basic example that well be using, is at the ncategory cafe here. We use the gysin sequence and lerayserre spectral sequence to aid in our computations. This last computation, as well as more details for the rest of the talk, can be found in hp, chapter 3.
Like most explanations of spectral sequences, it focuses on a particular spectral sequence. Consider a fiberoriented sphere bundle with total space e, base space m, fiber s k and projection map. We discuss the relations between the atiyahhirzebruch spectral sequence and the gysin map for a multiplicative cohomology theory, on spaces having the homotopy type of a finite cwcomplex. For example, in the serre spectral sequence for homology the. 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. Use the lerayserre spectral sequence for the homotopy bration cp1. An essay on spectral sequences university of washington. But we view the serre spectral sequence as a special case of the spectral sequence of a ltered space or ltered chain. Gysin map, which is a natural pushfoward in cohomology. Some extremely brief notes on the leray spectral sequence. Two typical examples of spectral sequence computations are reported. In contrast with the situation for the serre spectral sequence, for the eilenbergmoore spectral sequences the homology and cohomology versions arise in two different topological settings, although the two settingsareinasensedual.
Spectral sequence abstractsupplement definitions theorems. Xby means of he kunneth theorem, namely that there is a split short exact sequence 0. Group cohomology and the hochschildserre spectral sequence 12 10. As a sample application, we prove the gysin sequence and use it to compute h cpn. An essay on spectral sequences may 15, 2016 this is primarily an essay on the serre spectral sequence for the homologycohomology of a bration. There are two eilenbergmoore spectral sequences that we shall consider, one for homology and the other for cohomology. There were two large problem sets, and midterm and nal papers. It was introduced by gysin, and is generalized by the serre spectral sequence definition. This spectral sequence turns out to be an adams spectral sequence in the category of knlocal ehk nmodules. 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. Last time we defined cohomology in homotopy type theory. This case could also be deduced from the gysin sequence in 4. 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.
Introduction algebraic topology focuses on the properties of spaces and maps between. The maps forming chain complexes on each page are known as. William kronholm, the roggraded serre spectral sequence, homology homotopy appl. Fibrations and the leray serre spectral sequence 4 3. Adams spectral sequence, continuous homotopy xed point spectra, morava stabilizer group. We already know how to do this in case of the trivial bre sequence f. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx.
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. We can obtain this sequence from the serre spectral sequence if the base is simply connected, but the gysin sequence exists even for nonsimply connected base manifolds via the thom isomorphism. Symbolic computation, spectral sequences, serre spectral sequence, eilenbergmoore spectral sequence, constructive algebraic topology, common lisp. Fibrations and the lerayserre spectral sequence 4 3. Once weve pinned these down, we will also mention some of the most common complications and use. 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.
Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences. We also summarize numerous related computations, taken from mcclearys \a users guide to spectral sequences. There are essentially two situations in which spectral sequences arise. Gysin sequence and prove its exactness dk, theorem 9. It induces, and is induced by, the thom isomorphism. 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. 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. Gysin sequence and hochschildserre spectral sequence. Convergence proofs 12 references 15 this chapter contains those.
The serre spectral sequence and first applications. It expresses, in the language of homological algebra, the singular homology of the total space x of a fibration in terms of the homology of the base space b and the fiber f. In modern parlance, gysin had identi ed a form of the transgression homomorphism which, in the case of sphere spaces, is realized as a long exact sequence called the gysin sequence h pm. Introduction to spectral sequences ucb mathematics. Use then the serre spectral sequence to construct the gysin sequence and prove its exactness dk, theorem 9. A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. Spanier 10 that this sequence can be derived from the serre spectral sequence. The sequence may be obtained as a corollary of the serre spectral sequence for the given fibration. Gysin sequence of a circle bundle, which may be regarded as an immediate consequence of the serre spectral sequence for fiber bundles.
Spectral sequence abstractsupplement in this talk i plan to introduce spectral sequences by explaining how they arise from a ltration of an abelian group and then performing a couple of instructive applications of the lerayserre spectral sequence. The serre spectral sequence in cohomology ha2, section 5. I will also try to brie y touch the topics of local coe cients and convergence. It follows from 1 and 2 that the entry of every subsequent page of the spectral sequence is also zero. Spectral sequences in topology university of michigan. Chapter 1 spectral sequences in general b efore we start in on computations with spectral sequences, we should take a moment to outline what they are and where they come from. The string bracket turns out to be nonzero on torsion coming from string homology. 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. 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 erent named uses. Lagrangian intersections and the serre spectral sequence. Use the cohomological lerayserre spectral sequence and the derivation property to compute d 3 explicitly, using the cohomology ring of cp1. The result is due to jeanpierre serre in his doctoral dissertation.
It was introduced by gysin, and is generalized by the serre spectral sequence. 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. We find that over the integers, the string lie algebra bracket structure is more interesting, and not always zero as in basus thesis. Let ebe a spectral sequence, and suppose that for some r. Summary the serre spectral sequence e ectively computes cohomology rings for numerous classical spaces. For more nice explanations of spectral sequences, see 1 and 2. It explores how the spectral sequence introduced there interacts with the presence of bubbling. In mathematics, the serre spectral sequence is an important tool in algebraic topology. In the field of mathematics known as algebraic topology, the gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The question naturally arises therefore whether the serre spectral sequence can be generalized to the case of an arbitrary. The gysin sequence is a useful tool for calculating the cohomology rings given the euler class of the sphere bundle and vice versa. Alan hatcher, spectral sequences in algebraic topology i.
Jeanpierre serre, homologie singuliere des espaces fibres applications, ann. Gysin studied the homology structure of a sphere space composed of manifolds via a construction associated to a simplicial mapping. Group cohomology lecture notes university of washington. The spectral sequence converges to the singular homology of the topological spacee. However, this sequence is only able to compute the tdual curvature class in the case where the projection of the hflux to the e u00b9 u00b2u221e term of the lerayserre spectral sequence is trivial. Math 231br advanced algebraic topology taught by alexander kupers notes by dongryul kim spring 2018 this course was taught by alexander kupers in the spring of 2018, on tuesdays and thursdays from 10 to 11.
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