During world war ii hurewicz gave evidence of surprising versa. A contribution of a different nature was his extension of g. Instead of using the serre spectral sequence, we only assume the classical hurewicz theorem, and give a short proof of the rational gysin and. Lecture notes algebraic topology ii mathematics mit. From the long exact sequence of homotopy groups associated to x s3 kz,3 we get. In this survey, i present a compilation of the homotopy groups of tmf and of its various localizations. You can use without proof the fact that cohomology with rational coe cients is dual to homol. The lerayserre spectral sequence for the cohomology and the homology of a fibration is introduced and the e2 term is evaluated.
And always, one of the most mystifying things in algebraic topology to me was the notion of a spectral sequence. At 14 dec 2019 on the tmf hurewicz image of a1 vietcuong pham december 17, 2019 abstract let a1 be any spectrum in the class of. A space xis locally nite if h ix is nitely generated for each i. Torsion exponents in stable homotopy and the hurewicz.
F2 survives to the e1page of the adams spectral sequence, then its image under the hurewicz map. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians. The classical adams spectral sequence, in contrast, can be quickly set up and used to do some calculations which would be quite di. Indeed, a main application is the computation of various homotopy groups of spheres, and we will demonstrate this by the. Bockstein, grothendieck, hurewicz, kunneth, quillen, van kampen. On the structure of higher terms of the spectral sequence of a fibre space with e. The adamsnovikov spectral sequence and the homotopy groups. A variety of applications are given, including the hurewicz theorem in homotopy theory, and a calculation of.
Commutative ring spectra and spectral sequences robert bruner department of mathematics. This includes product spaces and covering spaces as special cases, and can be proven by the serre spectral sequence on homology of a fibration. These are very useful for calculating homologycohomology of topological spaces. I may unify them in the future, but since drawing spectral sequence is a timeconsuming task, i. What are the relations connecting the homology structure of the bundle, base space, ber and group. The topology of fiber bundles lecture notes ralph l. Indeed, a main application is the computation of various homotopy groups of spheres, and we will demonstrate this by the calculation of. As a consequence, we will derive some homotopytheoretic applications. There will also be some group cohomology calculations. Hurewicz images of real bordism theory and ern 3 to introduce the family, we appeal to lins complete classi cation of the groups ext 4. Samuel eilenberg 1949 it is now abundantly clear that the spectral sequence is one of the. The adams spectral sequence john rognes june 5th 2012 contents 1 stable homotopy theory 2.
Birkhoffsergodic theorem to spaces without invariant measure 42. Novikov spectral sequence is 1, although the point of view im adopting here is from the. Pdf a quick proof of the rational hurewicz theorem and a. Description of the spectral sequence the space cp has a natural filtration, namely by the subcomplexes cpp, and a natural basepoint cp. The idea behind the adams spectral sequence hereinafter know as the. Focus on ext, tor, and the serre and universal coe cient spectral sequences. In b1 we described a hurewicz spectral sequence going from homotopy to homology. Use then the serre spectral sequence to construct the gysin sequence and prove its. Hurewicz let x be a simply connected space, and let i. They have a reputation for being abstruse and difcult. They were introduced by leray in the 1940s at the same time as he introduced sheaves. He uses the fact that, up to that point, the only reliable way to compute homotopy groups was to compute homology groups and then apply the hurewicz theorem. The lefthand map is an isomorphism by the usual hurewicz theorem since f is n.
Robert bruner, an adams spectral sequence primer, 2009. Hurewicz made a number of excursions into analysis, principally real variables. Sometimes these are detailed, and sometimes they give references in the following texts. As a result youll see two ways of drawing spectral sequences in this document. We describe here, for any pointed connected space x and ring r, a firstquadrant spectral sequence, first proposed by h. Conversely, a serre fibration between cwcomplexes is a hurewicz fibration. For a g lie foliation, we give an isomorphism between the e 2 term of the spectral sequence and the reduced cohomology of g in the sense of st hu, 7 with coefficients in the foliated.
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. The readers should be warned that the may spectral sequence at the top of the diagram above is not the classical may spectral sequence. More sepci cally, the sinvariant sk of a knot kgives a lower bound on the slice 4ball genus of. In particular, the hurewicz theorem and the long exact sequence of homotopy imply that. Therefore, h x is an isomorphism mod csince the diagram commutes. There are also spectral sequences, which is an algebraic way to reassemble ltration quotients into the original object.
Spectral sequences are introduced in the general context of filtered chain and cochain complexes. Lecture notes on homotopy theory and applications math. From now on we shall say briefly that a given triangle 0. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. We cover essentially only that part of the theory needed in algebraic geometry. Since ris a pid, it follows from the universal coe. Use the serre spectral sequence and postnikov towers to conclude that if x is a 1connected space whose homotopy groups are all nite, then h x.
Further, the classical adams spectral sequence is still a useful calculational and theoretical tool, and is an excellent introduction to the general case. The equivariant dual steenrod algebra and adams spectral sequence 8. The adams spectral sequence is a vast generalization of the computation of homotopy groups from cohomology groups via the hurewicz theorem. Math 231br advanced algebraic topology lecture notes. On the differentials of the spectral sequence of a fibre bundle. Hurewicz s higher homotopy groups could be seen to be denumberable for a poly.
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. The bockstein and the adams spectral sequences article pdf available in proceedings of the american mathematical society 831 september 1981 with 26 reads how we measure reads. The homotopy groups of tmf and of its localizations. The spectral sequence let x e y and let r be a ring. The adams spectral sequence appeared in the 60s trying to determine the homotopy groups of some topological spaces. Adams filtration and generalized hurewicz maps for infinite. Bwith ber f, this spectral sequence relates the cohomology of the total space eto the cohomologies of band f. For any connected space x and ring r, we describe a firstquad. On the other hand, if n 1 mod 4 there is no pair of everywhere independent vector elds on sn.
We will rst cover the homotopy theory of spaces, generalized cohomology, stable homotopy theory, and spectral sequences. Get a printable copy pdf file of the complete article 534k. Fadell, international symposium on algebraic topology. The lectures were given in strasbourg, may 711, 2007. The spectral sequence whose existence is asserted in the above theorem is an example of a. The second map in 1 is the boundary homomorphism of the elliptic spectral sequence. 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. A fibration or hurewicz fibration or hurewicz fiber space, so named after witold hurewicz is a continuous mapping. May 21, 2007 contents 1 the adams spectral sequence 2 2 classical calculations 5 3 the adamsnovikov spectral sequence 10 4 complex oriented homology theories 5 the height. Contents 1 preface 2 2 the serre spectral sequence 4 3 spectral sequences in general 22 4 the spectral sequence of a ltered complex 28.
Robert bruner wayne state university ring spectra and spectral sequences ring spectra hamburg 22 48. Dyerlashof operations in homology under the hurewicz map. The spectral sequence now we can define the homotopy spectral sequence of a space x e. The homotopy groups of tmf and of its localizations andre henriques. Origins to 1953, in history of topology, 631663, edited by ioan james, northholland, amsterdam, 1999. Ravi vakil spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. 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. Aug 19, 20 this chapter is devoted to the lerayserre spectral sequence for the homology and cohomology of a fibration.
The hurewicz theorem and other applications using the serre spectral sequence, prove the classical hurewicz theorem bt, theorem 17. We will talk about an important spectral sequence in algebraic topology called the serre spectral sequence, which comes in both homological and cohomological varieties. This spectral sequence is called the adamsnovikov spectral sequence. Spectral sequences fabian hebestreit, achim krause, thomas nikolaus june 27, 2017 abstract these are notes for a course taught at the university of bonn in summer 2017. The transfer and the spectral sequence of a filtration. Kernel of hurewicz map using the spectral sequence of the. Fiber bundles over paracompact bases constitute important examples. John rognes, the adams spectral sequence following bruner 09, 2012. Alexander kupers, an introduction to the adams spectral sequence following rognes 12 paolo masulli, stable homotopy and the adams spectral sequence. The homotopy spectral sequence of a space with coefficients in a ring 81 obtained by defining 6.
Lecture notes assignments download course materials. Sergeraert spectral sequences with kenzo zurich, june 2012 2. G, whose e2term is isomorphic to certain derived functors of the halgebra of xi. Localizing the e2 page of the adams spectral sequence. Here is how serre used the lerayserre spectral sequence to compute homotopy groups, in the simplest example. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. This may not be the complete list of references from this article. However, in constructing the equivariant may spectral sequence, we ltered am f and m.
Serre spectral sequences are a powerful theoretical and computational tool with numerous applications to algebraic topology. Operations on resolutions and the reverse adams spectral. Pdf on the differentials of the spectral sequence of a. Miller, which converges strongly to the reduced homology of x with coefficients in r. Given a topological space x with known homology groups, we can consider the eilenbergmoore spectral sequence to compute h x. In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. This chapter is devoted to the lerayserre spectral sequence for the homology and cohomology of a fibration. Lecture notes were posted after most lectures, summarizing the contents of the lecture.
Bwith ber f, this spectral sequence relates the cohomology of the total space eto the cohomologies of. Mod c hurewicz, whitehead, cohomology spectral sequence. Adrian will lecture on how to build a spectral sequence from a ltration. The equivalence of simplicial and singular homology 128. Continuing, if n 8e 1 7 mod 8, then there are 7 independent vector elds on sn. The classical may spectral sequence is constructed from an increasing ltration of the dual steenrod algebra a. From homological perturbation to spectral sequences. On the tmfhurewicz image of a1 vietcuong pham december 17, 2019 abstract let a1 be any spectrum in the class of. The whitehead theorem and the hurewicz theorem request pdf. Introduction to spectral sequences ucb mathematics.
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