Traditionally the adamsnovikov spectral sequence has been a tool which has enabled the computation of generators and relations to describe homotopy groups. We characterize the 2line of the plocal adamsnovikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p. Realizability of the adamsnovikov spectral sequence for formal amodules tyler lawson abstract. Moreover in topology at odd primes, the adamsnovikov spectral sequence is a more e. Pdf the adamsnovikov spectral sequence and voevodskys. The adamsnovikov spectral sequence and the homotopy groups of spheres paul goerss.
Here a natural geometric description of the sequence is given in terms of cobordism theory and. Michael andrews and haynes miller october 24, 2017 abstract we calculate the localization of the motivic stable homotopy ring over c, con rming a conjecture of guillou and isaksen. Furthermore, we calculate the associated algebraic novikov spectral sequence converging to the 1line of the bp chromatic spectral sequence. Using the landweberaraki theory of real cobordism and realoriented spectra, we define a real analogue of the adamsnovikov spectral sequence. From this information we then prove a second theorem in classical topology, namely proposition 8. Abstractusing the landweberaraki theory of real cobordism and realoriented spectra, we define a real analogue of the adamsnovikov spectral sequence. Actually, we first compute the e 2term of the anss by the bss and then study the adamsnovikov differentials, and obtain e.
Localizations and the adams novikov spectral sequence lecture 30 april 17, 2010 throughout this lecture, we x a ring spectrum e. The e 1term e s,t 1 k of the spectral sequence is an ext group of bp. Adamsnovikov spectral sequence p2 adamsnovikov spectral sequence p2 adams spectral sequence anss p3 ass p3 anss p3 55 59. Some algebraic aspects of the adamsnovikov spectral. This is in contrast to the situation in the classical adams spectral sequence, in which these ele ments occur in filtration increasing with t. Localizations and the adamsnovikov spectral sequence. As the prime grows, so does the range of computation. Adamsnovikov spectral sequence p2 adamsnovikov spectral sequence p2. Pdf on the adamsnovikov spectral sequence and products. This is a new spectral sequence with a potentially calculable e2term. The algebra rapidly becomes prohibitively difficult. Section 8 discusses the motivic adamsnovikov spectral sequence. We continue working at p 2, using the abbreviations h y h y. The microstable adams novikov spectral sequence 11 then each element of ext.
Themotivic adamsnovikov spectralsequenceat odd c r sven. It has versions converging to either the z2equivariant or the nonequivariant stable 2stems. By corollary 5, the descent spectral sequence is isomorphic to the adamsnovikov spectral sequence. But there is a very simple xresolution of the sphere, used by the second author in rav, chapter 7.
Some algebraic aspects of the adamsnovikov spectral sequence, thesis, princeton university, 1974. More in detail, the adamsnovikov spectral sequence is a class of spectral sequences which converge to and hence are used to compute homotopy groups of connective spectra, hence in particular the stable homotopy groups of spheres. We give a similar characterization of the 1line, reinterpreting some earlier work of a. Pdf in the adams novikov spectral sequence one considers ext groups over the hopf algebroid. We show that the formal amodule adamsnovikov spectral sequence of ravenel does not naturally arise froma ltration on a map of spectra by examining the case a zi. We give the precise relationship between the congruences of modular forms introduced by the rst author with the qspectrum and the finvariant of the second author. Realoriented homotopy theory and an analogue of the adams. Our approach is via the motivic adamsnovikov spectral sequence. Adamsnovikov spectral sequence and the chromatic spectral sequence, which are based on complex cobordism and formal groups, have yielded a wealth of data 31. Translating our gradings into algebraic geometry, we get the following result. This result is also viewed as the calculation of a direct limit of localized modified adams spectral sequences converging to the homotopy of the v1 periodic sphere spectrum. For background on stable homotopy theory see introduction to stable homotopy theory. The boadams spectral sequence mark behrens notre dame joint with agnes beaudry uchicago, prasit bhattacharya notre dame, dominic culver notre dame, zhouli xu uchicago generalized adams spectral sequences. We will assume for simplicity that eis a structured ring spectrum.
Introduction to the adams spectral sequence in nlab. The chromatic spectral sequence was introduced by miller, ravenel, and wilson to compute the e 2term of the adamsnovikov spectral sequence for computing the stable homotopy groups of spheres. These notes were taken in the homotopy theory learning. Wilson in their work on the novikov spectral sequence from the point of view of the classical adams spectral sequence. Boris i botvinnik traditionally the adamsnovikov spectral sequence has been a tool which has enabled the computation of generators and relations to describe homotopy groups. Asdiscussed in 5, this entirely algebraic spectral sequence is isomorphic to the cmotivic adams spectral sequence for mmf then one can pull back or push forward di. In section8we construct a spectral system for a cube of brations, where one would usually apply eilenbergmoore spectral sequences successively. Theoretically, they enable a calculation of the stable homotopy groups but they have broader utility than this. Periodic phenomena in the adamsnovikov spectral sequence. E4page ofthe adamsnovikovspectral sequence this chart shows the adamsnovikov e4page through the 90stem, excluding the v1periodic elements.
Ext group the microstable adams novikov e 2term for the following reason. Localizations and the adamsnovikov spectral sequence lecture 30 april 17, 2010 throughout this lecture, we x a ring spectrum e. For each spectrum tm one can set up a chromatic spectral sequence as in rav86, chapter 5. Miller and others published periodic phenomena in the adamsnovikov spectral sequence.
Chapter 4 bptheory and the adamsnovikov spectral sequence in this chapter we turn to the main topic of this book, the anss. This thesis is concerned with the application of certain computational methods from stable algebraic topology in motivic homotopy theory over padic fields. These are notes for a five lecture series intended to uncover largescale phenomena in the homotopy groups of spheres using the adamsnovikov spectral sequence. O x is l nlocal for some n, so the desired result follows from the smash product theorem, which is equivalent to the statement that the adamsnovikov spectral sequence for a l. The adamsnovikov spectral sequence is the e eadams spectral sequence, for e e mu. On relations between 1lines of adamsnovikov spectral. The adamsnovikov spectral sequence and the landweber. May spectral sequence, adams spectral sequence, cohomology of the steenrod algebra, adamsnovikov spectral sequence the author was supported by nsf grant dms12022.
The microstable adamsnovikov spectral sequence 11 then each element of ext. Manifolds with singularities and the adamsnovikov spectral sequence. The adams novikov spectral sequence and the homotopy groups of spheres paul goerss. Adams spectral sequence many differentials differentials go back by 1 and up by r. A novices guide to the adamsnovikov spectral sequence. Pdf a novices guide to the adamsnovikov spectral sequence. Thus in the limit each s ngets replaced by an in nitesimal version of itself. This is a new spectral sequence with a potentially calculable e 2term.
The first line of the bockstein spectral sequence on a. We show that the spectral sequence converging to the stable homotopy groups of spheres, induced by the betti realization of the slice tower for the motivic sphere spectrum, agrees with the adamsnovikov spectral sequence, after a suitable reindexing. The iijterm of the classical adams spectral sequence hereafter abbreviated class is isomorphic to ext a z2, z2, where a is the mod 2 steenrod algebra. Conversely, there is a spectral sequence of adamsnovikov type.
It is natural to ask whether this map of spectral sequences arises as a. In section 1 we develop the basic properties of mu and the brownpeterson spectrum bp, using the calculation of trmu 3. The algebra rapidly becomes prohibitively difficult, and information at the e2 level will no longer suffice. For any spectrum x, the bordism groups mu evenx form a module over the lazard ring l. The adamsnovikov spectral sequence and the landweber exact functor theorem gabriel valenzuela april 22, 20 1 the adamsnovikov spectral sequence if eis a su ciently nice ring spectrum, theres a spectral sequence e 2 ext s. Download fulltext pdf on the adamsnovikov spectral sequence and products of. To any spectrum x, we can associate the cosimplicial ring spectrum n 7. Pdf spectral sequence that bears his name has been a source of fascination to homotopy theorists. I determine the coefficients of 2complete algebraic cobordism and a type of connective algebraic ktheory in the motivic. The proof relies on an extension of delignes d\ecalage construction to the tottower of a cosimplicial spectrum. The appendices contain charts of the motivic adams spectral sequence. The resulting adamsnovikov resolution of moh8iputs severe restrictions on its adamsnovikov spectral sequence. Each morava stabilizer group s n gets replaced by a certain open subgroup which shrinks asmincreases. On relations between 1lines of adamsnovikov spectral sequences modulo invariant prime ideals takeshi torii department of applied mathematics, fukuoka university, fukuoka 8140180, japan.
The reader will have noticed a clash of spectral sequence cultures between the slice and adamsnovikov spectral sequences. The adamsnovikov spectral sequence and the homotopy. This page gives a detailed introduction to the adams spectral sequence in its general spectral form adamsnovikov spectral sequence. It has versions converging to either the z 2equivariant or the nonequivariant stable 2stems. We present a detailed analysis of 2complete stable homotopy groups, both in the classical context and in the motivic context over c. My main tools are motivic analogues of the adams and adamsnovikov spectral sequences. Chapter 4 bptheory and the adamsnovikov spectral sequence.
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