Hans Joachim Baues, Mamuka Jibladze
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.
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Quivers and the Adams spectral sequence
本文提出一种基于箭图表示同调代数识别 Adams 谱序列 $E_2$-项的新方法，其适用范围广于传统下降平坦性技术，并在 $p$-局部整同调情形下给出了基于经典 Adams 谱序列的 $E_2$-项完全分解。
The Borel and genuine $C_2$-equivariant Adams spectral sequences
本文建立了截断实射影谱的经典 Adams 谱序列与 2-完备球面的 Borel 和 genuine C_2-等变 Adams 谱序列之间的联系，从而将 genuine 等变谱序列的计算归结为可计算的 Borel 等变谱序列。
Adams spectral sequences and Franke's algebraicity conjecture
本文通过为表现良好的同调理论构造导出 ∞-范畴来编码 Adams 谱序列，并以此证明 Franke 代数性猜想及 Adams 滤过的同伦相干幺半性。
We describe an algorithm for the automated deduction of many $d_2$ differentials in the Adams spectral sequence. We discuss our implementation and the results of the computation.
We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property.
We study the motivic Adams-Novikov spectral sequence at an odd prime $l$ over the base fields $\mathbb{C}$ and $\mathbb{R}$. This spectral sequence converges to the stable motivic homotopy groups of the $l$-completed motivic sphere spectrum. We show that the spectral sequence is determined by the topological Adams-Novikov sequence, similar to the situation at the prime two over the base field $\mathbb{C}$.
The purpose of this short article is to record the computation of the homotopy groups of 3-local $\mathrm{tmf}$ via the Adams spectral sequence.
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
We describe a variant construction of the unstable Adams spectral the sequence for a space $Y$, associated to any free simplicial resolution of $H^*(Y;R)$ for $R=\mathbb{F}_p$ or $\mathbb{Q}$. We use this construction to describe the differentials and filtration in the spectral sequence in terms of appropriate systems of higher cohomology operations.
We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic Adams spectral sequence is not strongly convergent over number fields. As applications we give bounds on the exponents of the $(\ell,η)$-completed motivic stable stems, and calculate the zeroth $(\ell,η)$-completed motivic stable stems.
We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property.
We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the $d_2$-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-$J$ spectrum, finishing a calculation that was begun by D. Davis in 1975.
We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by $\mathbb{F}_2$-synthetic spectra.
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.
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.
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.
In this paper we establish a formula for computing $d_2(sq^i(x))$ where $x$ is a permanent cycle in the $C_2$-equivariant Adams spectral sequence or the motivic Adams spectral sequence over $Spec(\mathbb{R})$. This requires establishing that the Adams towers have an $H_{\infty}$-structure as well as determining the attaching maps for $C_2$-equivariant projective spaces. The attaching maps of $C_2$-equivariant projective spaces can then be used to determine the coefficients of differentials in both the equivariant and motivic case. At the end some sample computations are given.
We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology is proved in two new cases.
Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the E_1 term of the E-Adams spectral sequence. The main theorems of this paper concern when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to c…
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.
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.
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 use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.
This paper is an English translation of chapter nine of the book "Adams spectral sequence and stable homotopy groups of spheres" by Jinkun Lin (in Chinese)(Sciences Press, Beijing 2007). In this paper, a sequence of new indecomposable families in the stable homotopy groups of spheres such as h_0h_n,h_nb_n,h_0h_nh_m,h_0(h_mb_{n-1}-b_{m-1}h_n) families was detected. The proof of all the detections was stated in this paper in based on several published papers by the author.
We show that there is a non-trivial differential $d_4(h_6^3) =h_0^3g_4$ in the mod 2 Adams spectral sequence for spheres. This together with the results in \cite{barratt_differentials_1970,lin_differential_1998,kan_differential_2001} completely settle the differentials of $h_i^3$ for $i\ge4$. (The differentials of $h_i^3$ for $i=0,1,2,3$ are well-known.) Our proof uses the Kevaire invariant elements $θ_i \inπ_{2^{i+1}-2}^S$ for $i=4,5$ with the properties $2θ_4 =0$, $2θ_5 =0$.
Let $E=E_n$ be Morava $E$-theory of height $n$. In previous work Devinatz and Hopkins introduced the $K(n)$-local $E_n$-Adams spectral sequence and showed that, under certain conditions, the $E_2$-term of this spectral sequence can be identified with continuous group cohomology. We work with the category of $L$-complete $E_*E$-comodules, and show that in a number of cases the $E_2$-term of the above spectral sequence can be computed by a relative Ext group in this category. We give suitable conditions for when we can identify this Ext group with continuous group cohomology.
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 …
The $\mathbb{C}$-motivic Adams Spectral Sequence for $s\leq5$
本文利用 Burklund-Xu 技术确定了 C-动机 Adams 谱序列 E2 页前 5 行的 τ^n-挠元，证明该范围内元素要么是 τ^1-挠元要么是 τ-自由元，并给出 τ^n-挠元出现所需 Adams 滤过下界及可能的 3n 界证据。
Stable comodule deformations and the synthetic Adams-Novikov spectral sequence
本文研究合成谱中的 Adams–Novikov 谱序列，通过稳定余模形变识别 $\mathrm{E}_2$ 页并范畴化已有代数结果，且在 $p=2$ 时将球面计算推进至 45-stem。
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $π_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem. The proof of this result uses the $RP^\infty$ technique, which was introduced by the authors in \cite{WX1} to prove $π_{61}=0$. This paper advertises this method through examples that have simpler proofs than in \cite{WX1}.
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.
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 …
The $\mathbb C$-motivic Adams-Novikov spectral sequence for topological modular forms
本文分析了 $\mathbb{C}$-motivic 模形式谱 $\mathit{mmf}$ 及经典拓扑模形式谱 $\mathit{tmf}$ 的 Adams-Novikov 谱序列，主要采用纯代数技术，并解决了 $\mathit{tmf}$ 同伦群乘法结构的一个遗留细节。
2-track algebras and the Adams spectral sequence
本文引入2-track代数和三次链复形，展示了Adams谱序列的$E_4$项为三次Ext群，扩展了先前工作并提高了计算的可行性。
Some nontrivial secondary Adams differentials on the fourth line
本文利用二阶 Adams 微分与二阶代数 Novikov 微分的对应关系，计算了奇素数 $p \geq 5$ 时 Adams 谱序列第四线上的四族非平凡二阶微分，并恢复了前三线的所有二阶微分。
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's, and show that they are self-dual. Our main result is that the Adams differential $d_r$ in any Adams spectral sequence can be expressed as an $(r+1)$-fold Toda bracket and as an $r^{\text{th}}$ order cohomology operation. We also show how the …
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 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 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…
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 …
Adams spectral sequences for non-vector-bundle Thom spectra
针对传统Adams谱序列计算扭曲$R$-同调群需扭曲来自向量丛的限制，研究利用Baker-Lazarev的$R$-模Adams谱序列，对多类谱的大量扭曲计算$E_2$页，证明该假设不必要，并给出超引力反常抵消相关的计算实例。
Localized at almost all primes, we describe the structure of differentials in several important spectral sequences that compute the cohomology of classifying spaces of topological Kac-Moody groups. In particular, we show that for all but a finite set of primes, these spectral sequences collapse and that there are no additive extension problems. We also describe some appealing consequences of these results. The main tool is the use of the naturality properties of unstable Adams operations on these classifying spaces.
A Toda bracket convergence theorem for multiplicative spectral sequences
该研究在对称幺模稳定拓扑模型范畴中推广了联系Massey乘积与Toda括号的Moss定理，使其适用于幺模相容滤过导出的乘性谱序列，可应用于motivic相关谱序列的计算。
The algebraic Novikov spectral sequence for topological modular forms
本文研究拓扑模形式的代数Novikov谱序列，通过计算mmf/τ的C-动机Adams谱序列并分析τ乘法长正合列中的映射，以纯代数方法独立得到其Adams-Novikov谱序列E2页。
A correspondence between higher Adams differentials and higher algebraic Novikov differentials at odd primes
本文基于“τ的余纤维哲学”，证明了在奇素数处经典Adams谱序列的高阶微分在特定范围内与相应的高阶代数Novikov微分相一致。
Automated proofs of unstable Adams differentials
本文提出一种计算机辅助方法，利用不稳定莱布尼茨法则和EHP序列自然性计算不稳定Adams谱序列微分，给出球面2-主不稳定同伦群至50-茎的阶数界及证明示例。
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.
The Adams differentials on the $e$-family
本文利用Burklund-Xu谱序列证明了Adams 4线上的e-族新末日猜想。
We establish a differential $d_2(D_1)=h_0^2h_3g_2$ in the $51$-stem of the Adams spectral sequence at the prime $2$, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the $n$-stems for $n<60$ [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.
We calculate the $η$-localization of the motivic stable homotopy ring over the complex numbers, confirming a conjecture of Guillou and Isaksen. Our approach is via the motivic Adams-Novikov spectral sequence. In fact, work of Hu, Kriz, and Ormsby implies that it suffices to compute the corresponding localization of the classical Adams-Novikov $E_2$-term, and this is what we do. Guillou and Isaksen also proposed a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov.
A C-motivic modular forms spectrum mmf has recently been constructed. This article presents detailed computational information on the Adams spectral sequence for mmf. This information is essential for computing with the C-motivic and classical Adams spectral sequences that compute the C-motivic and classical stable homotopy groups of spheres.
As a step towards understanding the $\mathrm{tmf}$-based Adams spectral sequence, we compute the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$, using a small presentation of $L_{K(1)}\mathrm{tmf}$ due to Hopkins. We also describe the $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence.
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].
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 use an Adams spectral sequence to calculate the R-motivic stable homotopy groups after inverting eta. The first step is to apply a Bockstein spectral sequence in order to obtain h_1-inverted R-motivic Ext groups, which serve as the input to the eta-inverted R-motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor-Witt (4k-1)-stem has order 2^{u+1}, where u is the 2-adic valuation of 4k. This answer is reminiscent of the classical image of J. We also explore some of the Toda bracket structure of the eta-inverted R-motivic stabl…
The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4
本研究通过 Adams 谱序列计算分类空间连通实 k-同调，结合两类α不变量核检测定理，证明了Z/4×Z/4群的Gromov-Lawson-Rosenberg猜想，同时确定了相关谱序列微分并研究了对应子Hopf代数的谱序列上积结构。
Algebraic redshift in the $C_2$-equivariant Adams spectral sequence
研究C₂等变稳定同调中v_n周期现象，通过C₂等变Adams谱序列在素数2处的视角，探讨相关类在某些有限生成子代数的上同调中的构造与检测，定义v_n周期或v_n torsion类，并展示染色过滤。
Machine Proofs for Adams Differentials and Extension Problems among CW Spectra
本研究通过计算方法获取大量Adams微分与扩张，构建含49个CW谱等信息的Zenodo数据集并提供机器证明，结合人类特设论证解决了126维最后的凯瓦雷不变量问题。
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.
The motivic lambda algebra and motivic Hopf invariant one problem
本文引入动机 lambda 代数计算动机 Adams 谱序列，据此分类含单位乘积的动机球面，并确定多类基域 1-线上的微分与永久循环。
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