The $E_2$-term of the Adams spectral sequence for $\mathbf{Y}$ may be described in terms of its cohomology $E^\ast \mathbf{Y}$, together with the action of the primary operations $E^\ast \mathbf{E}$ on it, for ring spectra such as $\mathbf{E} = \mathbf{H}\mathbb{F}_p$. We show how the higher terms of the spectral sequence can be similarly described in terms of the higher order truncated $\mathbf{E}$-mapping algebra for $\mathbf{Y}$ $\; - \;$ that is truncations of the function spectra $\operatorname{Fun}(\mathbf{Y}, \mathbf{M})$ for various $\mathbf{E}$-modules $\mathbf{M}$, equipped with the action of $\operatorname{Fun}(\mathbf{M}, \mathbf{M}')$ on them.
The aim of this paper is to gain explicit information about the multiplicative structure of l_*l, where l is the connective Adams summand. Our approach differs from Kane's or Lellmann's because our main technical tool is the MU-based Kuenneth spectral sequence. We prove that the algebra structure on l_*l is inherited from the multiplication on a Koszul resolution of l_*BP.
In recent work by Clarke, Crossley and the second author, various algebras of stable degree zero operations in p-local K-theory were described explicitly. The elements are certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for p-local cobordism and for BP, thus identifying the "Adams subalgebra" of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations.
The Adams spectral sequence was invented by J.F.Adams almost fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to solve this problem in the case of Z/2 coefficients and find inductive formulas for the differentials. It is based on the Stasheff algebra structures, operad methods and functional homology operations.
We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.
We modify a classical construction of Bousfield and Kan to define the Adams tower of a simplicial nonunital commutative algebra over a field k. We relate this construction to Radulescu-Banu's cosimplicial resolution, and prove that all connected simplicial algebras are complete with respect to André-Quillen homology. This is a convergence result for the unstable Adams spectral sequence for commutative algebras over k.
A classical theorem of Adams, Harris, and Switzer states that the 0th grading of complex $K$-theory cooperations, $KU_0ku$ is isomorphic to the space of numerical polynomials. The space of numerical polynomials has a basis provided by the binomial coefficient polynomials, which gives a basis of $KU_0ku$. In this paper, we produce a new $p$-local basis for $KU_0ku_{(p)}$ using the Adams splitting. This basis is established by using well known formulas for the Hazewinkel generators. For $p=2$, we show that this new basis coincides with the classical basis modulo higher Adams filtration.
We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field. We discuss several features of the associated Adams spectral sequence, including the basic construction and convergence properties. The paper also deals with motivic versions of the May and Adams-Novikov spectral sequences. It is shown how these tools can be used to give new proofs of some classical results in algebraic topology. Also, the considerations reveal the existence of certain "exotic" motivic homotopy classes which have no classical analogues.
The Adams spectral sequence was invented by J.F.Adams fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the most difficult problem of Algebraic Topology. Here we consider an approach to find inductive formulas for the differentials. It is based on the A_\infty-structures, E_\infty-structures and functional homology operations. As This approach it will be applied to the Kervaire invariant problem.
Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain complexes, resolutions, and derived functors. We show that higher order resolutions exist in higher track categories, and that they determine higher order Ext-groups. In particular, the E_m-term of the Adams spectral sequence (m<n+3) is a higher order Ext-group, which is determined by the track algebra of higher cohomology operations.
We compute the Adams-Novikov E_2-term of a spectrum Q(2) constructed by Behrens. The homotopy groups of Q(2) are closely tied to the 3-primary stable homotopy groups of spheres; in particular, they are conjectured to detect the homotopy beta family of Greek letter elements at the prime 3. Our computation leverages techniques used by Behrens to compute the rational homotopy of Q(2), and leads to a conjecture that the Adams-Novikov E_2-term for Q(2) detects the algebraic beta family in the BP-based Adams-Novikov E_2-term for the 3-local sphere.
In the early 2000's, Baues computed the secondary Steenrod algebra, the algebra of all secondary cohomology operations. Together with Jibladze, they showed that this gives an algorithm that computes all Adams $d_2$ differentials for the sphere. The goal of this paper is to reinterpret their results in the language of synthetic spectra in order to achieve stronger computational results. Using this, we obtain an algorithm that computes hidden extensions on the $E_3$ page that jump by one filtration, in addition to the $d_2$ differentials of Baues--Jibladze. We then implement and run this algorit…
In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $\mathbb{R}P^\infty$ with the aid of a computer, which gives us its Adams $E_2$-page in the range of $t<72$. We also compute the transfer map on the Adams $E_2$-pages. These data are used in our computations of the stable homotopy groups of $\mathbb{R}P^\infty$ in [6] and of the stable homotopy groups of spheres in [7].
An algorithm is described giving effective determination of the second differential in the Adams spectral sequence. The algorithm is based on the notion of secondary derived functor, and on the explicit algebraic model of the groupoid enriched category of stable maps and stable tracks between Eilenberg-Mac Lane spaces.
Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of ``additive groupoid enriched categories'', in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E_3-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E_3-term explicitly by an algorithm.
The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the …
We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory.
We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based Adams resolution is compatible with the augmentation ideal filtration of the range associated to the suspension spectrum of X(0)_+, an augmented commutative S-algebra. The proof makes use of the interplay of this filtration with Topological Andre Quillen Homology. An application is a Connectivity Theorem: Localize at a prime p and suppose X is (c-1)-connect…
In previous work it is shown that there is an abelian category A(G) constructed to model rational G-equivariant cohomology theories, where G is a torus of rank r together with a homology functor \piA_* : Gspectra ---> A(G), and an Adams spectral sequence Ext_{A (G)} (\piA_*(X), \piA_*(Y)) ===> [X,Y]^G_* In joint work with Shipley (arxiv:1101.2511), it is shown that the Adams spectral sequence can be lifted to a Quillen equivalence Rational-Gspectra = DG-A (G). The purpose of the present paper is to prove that A(G) has injective dimension precisely r, and to construct certain torsion functors a…
There is only one nontrivial localization of $π_*S_{(p)}$ (the chromatic localization at $v_0=p$), but there are infinitely many nontrivial localizations of the Adams $E_2$ page for the sphere. The first non-nilpotent element in the $E_2$ page after $v_0$ is $b_{10}\in \mathrm{Ext}_A^{2p(p-1)-2}(\mathbb{F}_p,\mathbb{F}_p)$. We work at $p=3$ and study $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand $\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ of $\mathrm{Ext}_A(\mathbb{F}_3,\mathbb{F}_3)$ above a lin…
We compute the h_1-localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the eta-local motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.
We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, eta, and nu, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy …
The James fibrations give rise to the geometric EHP sequences of homotopy groups of spheres. Using techniques from the Lambda algebra, \cite{BCKQRS66} shows that there are similar long exact sequences of Ext groups defining the $E_{2}-$page of the Bousfield-Kan spectral sequence (also known as the unstable Adams spectral sequence) computing homotopy groups of spheres. In this paper, we give another proof for this phenomenon.
Let $Φ\to Γ\to Σ$ be a conormal extension of Hopf algebras over a commutative ring $k$, and let $M$ be a $Γ$-comodule. The Cartan-Eilenberg spectral sequence $$ E_2 = \mathrm{Ext}_Φ(k,\mathrm{Ext}_Σ(k,M)) \implies \mathrm{Ext}_Γ(k,M)$$ is a standard tool for computing the Hopf algebra cohomology of $Γ$ with coefficients in $M$ in terms of the cohomology of the pieces $Φ$ and $Σ$. Bruner and Rognes, generalizing a construction of Davis and Mahowald, have introduced a generalization of the Cartan-Eilenberg spectral sequence converging to $\mathrm{Ext}_Γ(k,M)$ that can be defined when $Φ= Γ\squar…
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E. In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum. We show that the Adams Spectral Sequence for S_R based on a commu…
In this paper, we study the algebraic cobordism spectrum $MSL$ in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field $k$. Using the motivic Adams spectral sequence, we compute the geometric part of the $η$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\neq2, char k$). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to $MSL$. We determine its image, and identify its kernel as the ideal generated by differences of $SL$-flops. This was proved by B. Totaro in the complex analyt…
We calculate the higher homotopy groups of the Deligne-Getzler infinity-groupoid associated to a nilpotent L-infinity algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.
For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $\mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $ρ$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Mo…
In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding \emph{Maurer-Cartan elements} are connected by an algebra autom…
The focus of this paper is the comparison of two unstable homotopy spectral sequences-- the unstable mod p Adams spectral sequence that computes the unstable homotopy of a p-complete space, and the Goerss--Hopkins spectral sequence, which computes the unstable homotopy of the space of E-infinity maps between Hk-algebras, where k is the algebraic closure of the field with p elements and p is an odd prime. Using an adjunction between p-complete nilpotent spaces and a subset of Hk-algebras, this paper shows that the unit of this adjunction provides an isomorphism between these spectral sequences.
We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL<n> over p-adic fields. These spectra interpolate between integral motivic cohomology (n=0), a connective version of algebraic K-theory (n=1), and the algebraic Brown-Peterson spectrum. We deduce that, over p-adic fields, the 2-complete BPGL<n> split over 2-complete BPGL<0>, implying that the slice spectral sequence for BPGL collapses. This is the first in a series of two papers investigating motivic invariants of p-adic fields, a…
Let $C$ be a differential graded coalgebra, $ \barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.
Pure braid groups and pure mapping class groups of a punctured sphere have many features in common. In the paper the graded Lie algebra of the descending central series of the pure mapping class of a sphere is studied. A simple presentation of this Lie algebra is obtained.
We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.
A covariant functor from the category of mapping tori to a category of AF-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding AF-algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension 2, 3 and 4.
For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/τ$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for…
Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq^1, Sq^2, and Sq^4. The method of calculation is a motivic version of the May spectral sequence. Speculatively assuming that there is a "motivic modular forms" spectrum with certain properties, we use an Adams-Novikov spectral sequence to compute the homotopy of such a spectrum at the prime 2.
We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic interpretation, where the domain of the morphism is the shifted Hochschild cocohain complex. We then use this period mapping to give some applications such as unobstructedness of deformations.
The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions…
In this paper, we give a concrete description of the higher homotopy groups (n>0) of the mapping space Map_{Alg}(R,S) for R and S unbounded differential graded algebras (DGA) over a commutative ring k. In the connective case, we describe the relation between the higher (negative) Hochschild cohomology $HH^{-n+1}(R,S)$ and higher homotopy groups $π_{n} Map_{Alg}(R,S)$, when $n>1$.
