On Sikora's spectral sequences

Borel 与 genuine C_2-等变 Adams 谱序列

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。

Higher spectral sequences

In this article we construct what we call a higher spectral sequence for any chain complex (or topological space) that is filtered in $n$ compatible ways. For this we extend the previous spectral system construction of the author, and we show that it admits considerably more differentials than what was previously known. As a result, this endows the successive Leray--Serre, Grothendieck, chromatic--Adams--Novikov, and Eilenberg--Moore spectral sequences of the author with the structure of a higher spectral sequence. Another application is a universal coefficient theorem analog for spectral sequ…

The motivic Adams-Novikov spectral sequence at odd primes over $\mathbb{C}$ and $\mathbb{R}$

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 $\mathbb{C}$.

Localization of $(\infty, 1)$-categories and spectral sequences

We describe two types of localization for $(\infty, 1)$-categories which determine the successive terms in the homotopy spectral sequence of a (co)simplicial object.

Multiplicative structures on homotopy spectral sequences I

This mostly expository paper records some basic facts about towers of homotopy fiber sequences. We give a proof that a pairing of towers induces a pairing of associated spectral sequences, for towers of spaces and towers of spectra.

谱序列，décalage，与Beilinson t-结构

Spectral sequences, décalage, and the Beilinson t-structure

本文通过décalage和Beilinson t-结构解释了谱序列理论。

The T-algebra spectral sequence: Comparisons and applications

In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence in detail. Under certain assumptions, we show that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence agree. Under further assumptions, we can apply a variation of an argument due to Jennifer French and show that these spectral sequences agree with the unstable Adams spectral sequence. From these equivalences we obtain i…

The Bracket in the Bar Spectral Sequence for an Iterated Loop Space

When $X$ is an associative H-space, the bar spectral sequence computes the homology of the delooping, $H_{*}(BX)$. If $X$ is an $n$-fold loop space for $n\geq2$ this is a spectral sequence of Hopf algebras. Using machinery by Sugawara and Clark, we show that the spectral sequence filtration respects the Browder bracket structure on $H_{*}(BX)$, and so it is moreover a spectral sequence of Poisson algebras. Through the bracket on the spectral sequence, we establish a connection between the degree $n-1$ bracket on $H_{*}(X)$ and the degree $n-2$ bracket on $H_{*}(BX)$. This generalizes a result …

The RO(G)-Graded Serre Spectral Sequence

In this paper the Serre spectral sequence of Moerdijk and Svensson is extended from Bredon cohomology to RO(G)-graded cohomology for finite groups G. Special attention is paid to the case G=Z/2 where the spectral sequence is used to compute the cohomology of certain projective bundles and loop spaces.

Automated differential computation in the Adams spectral sequence

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.

Equivariant Universal Coefficient and Kunneth Spectral Sequences

We construct hyper-homology spectral sequences of Z-graded and ROG-graded Mackey functors for Ext and Tor over G-equivariant S-algebras (A-infty ring spectra) for finite groups G. These specialize to universal coefficient and Kunneth spectral sequences.

A colocalization spectral sequence

Colocalization is a right adjoint to the inclusion of a subcategory. Given a ring-spectrum R, one would like a spectral sequence which connects a given colocalization in the derived category of R-modules and an appropriate colocalization in the derived category of graded modules over the graded ring of homotopy groups of R. We show that, under suitable conditions, such a spectral sequence exists. This generalizes Greenlees' local-cohomology spectral sequence. The colocalization spectral sequence introduced here is associated with a localization spectral sequence, which is shown to be universal…

Leray Spectral Sequence for Simplicial Maps

The Leray spectral sequence of a map $f$ computes the homology of the domain of $f$ from the fibers of $f$. In this expository paper, we relate in full detail the Leray spectral sequence associated to a simplicial map $f$ to the Leray cosheaves of $f$. We then give applications to level set persistent homology and Reeb spaces.

SeqSee：一种基于模式的谱序列可视化方法

SeqSee: A schema-based approach to spectral sequence visualization

SeqSee是一种基于模式的谱序列可视化软件系统，通过引入标准化JSON模式作为中间表示，将谱序列的数学计算与可视化解耦，并在经典和C-motivic Adams谱序列案例中验证了其有效性。

The slice spectral sequence for the cyclic group of order p

We describe the slice tower and slice spectral sequence for arbitrary suspensions of the Eilenberg-MacLane spectrum of an arbitrary Mackey functor for the cyclic group of prime order.

A spectral sequence for string cohomology

Let X be a 1-connected space with free loop space LX. We introduce two spectral sequences converging towards H^*(LX;Z/p) and H^*((LX)_hT;Z/p). The E2-terms are certain non Abelian derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.

Higher structure in the unstable Adams spectral sequence

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.

A comparison of spectral sequences computing unstable homotopy groups of $p$-complete, nilpotent spaces

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.

Spectral sequence operations converge to Araki-Kudo operations

Previously we constructed operations in the mod 2 homology spectral sequence associated to a cosimplicial E-infinity space X. The correct target for this spectral sequence is the homology of Tot X. Noting that in this setting Tot X is an E-infinity space, we show that our operations agree with the usual Araki-Kudo operations in the target. We also prove that the multiplication in the spectral sequence agrees with the multiplication in H_*(Tot X).

The loop-coproduct spectral sequence

Let $M$ be a closed oriented $d$-dimensionnal manifold and let $LM$ be the space of free loops on $M$. In this paper, we give a geometrical interpretation of the loop coproduct and we study it's compatibility with the Serre spectral sequence associated to the fibration $ΩM \to LM \to M$. Then, we show that the spectral sequence associated to the free loop fibration $LN \to LX \to LM$ of some Serre fibration $N \to X \to M$ is a spectral sequence of Frobenius algebra.

The Adams spectral sequence for 3-local $\mathrm{tmf}$

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.

On the E^1-term of the gravity spectral sequence

The author constructed a spectral sequence strongly converging to h_*(Omega^n Sigma^n X) for any homology theory in [Topology 33 (1994) 631-662]. In this note, we prove that the E^1-term of the spectral sequence is isomorphic to the cobar construction, and hence the spectral sequence is isomorphic to the classical cobar-type Eilenberg-Moore spectral sequence based on the geometric cobar construction from the E^1-term. Similar arguments can be also applied to its variants constructed in [Contemp Math 293 (2002) 299-329].

Vanishing lines in generalized Adams spectral sequences are generic

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.

范畴正则扩张的LHS-谱序列

LHS-spectral sequences for regular extensions of categories

该研究将Xu提出的小范畴靶正则扩张LHS-谱序列推广至ext-群，构造源正则扩张的同类谱序列，得到三类Slominska谱序列特例，并证明多类经典谱序列可作为扩张的LHS-谱序列导出。

The local cohomology spectral sequence for topological modular forms

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald-Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

The Adams spectral sequence for the image-of-$J$ spectrum

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.

持久的勒雷谱序列

Persistent Leray's spectral sequence

本文构建了持久化的勒雷谱序列，通过空间的开集及其覆盖的交集计算持久同调。

Equivariant Spectral Sequences for Local Coefficients

We recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences and give a worked example of how to apply them in a computation involving the equivariant Serre spectral sequence. This paper contains some of the material in the author's Ph.D. thesis, which also discusses some results of L. Gaunce Lewis on the cohomology of complex projective spaces and corrects some flaws in his paper.

Leibniz rule on higher pages of unstable spectral sequences

A natural composition $\odot$ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for $p$-lower central series spectral sequence of a simplicial group. It is proved that $r$th differential satisfies a "Leibiz rule with suspension": $d^r(a\odot σb)=\pm d^ra\odot b+a\odot d^rσb,$ where $σ$ is the suspension homomorphism.

A spectral sequence in Bredon cohomology

For any finite group G, we construct a spectral sequence for computing the Bredon cohomology of a G-CW complex X, starting with the cohomology of X^H/\cup_{K>H}X^K with suitable local coefficients, for various H \leq G.

The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres

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.

Operations in the homology spectral sequence of a cosimplicial infinite loop space

Consider the mod 2 homology spectral sequence associated to a cosimplicial space X. We construct external operations whose target is the spectral sequence associated to EΣ_2 \times_{Σ_2} (X\times X). If X is a cosimplicial E_\infty-space, we couple these external operations with the structure map EΣ_2 \times_{Σ_2} (X\times X) \to X to produce internal operations in the spectral sequence. In the sequel we show that they agree with the usual Araki-Kudo operations on the abutment H_*(Tot X).

On the Slice Spectral Sequence

We introduce a variant of the slice spectral sequence which uses only regular slice cells, and state the precise relationship between the two spectral sequences. We analyze how the slice filtration of an equivariant spectrum that is concentrated over a normal subgroup is related to the slice filtration of its geometric fixed points, and use this to prove a conjecture of Hill on the slice filtration of an Eilenberg MacLane spectrum. We also show how the (co)connectivity of a spectrum results in the (co)connectivity of its slice tower, demonstrating the "efficiency" of the slice spectral sequenc…

Computing higher Leray-Serre spectral sequences of towers of fibrations

The higher Leray-Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray-Serre spectral sequence of a fibration. In this work, we present algorithms to compute higher Leray-Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray-…

Cellular approximations and the Eilenberg-Moore spectral-sequence

We set up machinery for recognizing k-cellular modules and k-cellular approximations, where k is an R-module and R is either a ring or a ring-spectrum. Using this machinery we can identify the target of the Eilenberg-Moore cohomology spectral sequence for a fibration in various cases. In this manner we get new proofs for known results concerning the Eilenberg-Moore spectral sequence and generalize another result.

Strong convergence in the motivic Adams spectral sequence

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.

The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differ…

Correction to: "A spectral sequence for stratified spaces and configuration spaces of points"

We correct some oversights in the paper "A spectral sequence for stratified spaces and configuration spaces of points" by the second named author. In particular we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.

拓扑模形式之代数Novikov谱序列

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。

The Arone-Goodwillie spectral sequence for $Σ^{\infty}Ω^n$ and topological realization at odd primes

We employ the Goodwillie spectral sequence for the iterated loop space functor in order to provide realizability conditions on certain unstable modules over the Steenrod algebra at an odd prime.

Spectral sequences of operad algebras

We identify conditions under which it is guaranteed that an action of an operad on the $E^2$ page of a spectral sequence passes to $E^r$ for $r\ge 2$ and hence to the $E^\infty$ page. We consider this question in both the purely algebraic and topological settings.

An extension in the Adams spectral sequence in dimension 54

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.

Stems and Spectral Sequences

We introduce the category Pstem[n] of n-stems, with a functor P[n] from spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of a space. We show how to associate to each simplicial n-stem Q an (n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem of a simplicial space X, the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X. Similar results are also proven for cosimplicial n-stems. They are helpful for computations, since n-stems in low degrees have good algebraic models.

E_\infty-structure and differentials of the Adams spectral sequence

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.

Stasheff structures and differentials of the Adams spectral sequence

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.

On the May Spectral Sequence at the prime 2

We make a conjecture about the whole $E_2$ page of the May spectral sequence in terms of generators and relations and we prove it in a subalgebra which covers a large range of dimensions. We show that the $E_2$ page plays a universal role in the study of Massey products in commutative DGAs. We conjecture that the $E_2$ page is nilpotent free and also prove it in this subalgebra. We compute all the $d_2$ differentials of the generators in the conjecture and construct maps of spectral sequences which allow us to explore Adams vanishing line theorem to compute differentials in the May spectral se…

On the Brun spectral sequence for topological Hochschild homology

We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the $E$-homology of $THH(A;B)$, where $E$ is a ring spectrum, $A$ is a commutative $S$-algebra and $B$ is a connective commutative $A$-algebra. The input of the spectral sequence are the topological Hochschild homology groups of $B$ with coefficients in the $E$-homology groups of $B \wedge_A B$. The mod $p$ and $v_1$ topological Hochschild homology of connective complex $K$-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We p…

持续同调与谱序列的分解

Decomposition of Persistent Homology and Spectral Sequences

研究域上过滤链复形的持续同调与谱序列关系，提出基于持续同调分解的方法，证明在一般假设下两者捕获相同信息。

Hopf algebras up to homotopy and the Bockstein spectral sequence

Anick proved that every q-mild Hopf algebra up to homotopy is isomorphic to a primitively-generated chain Hopf algebra. We provide a new proof, that involves extensive use of the Bockstein spectral sequence.

Spectral sequences via examples

These are lecture notes for a short course about spectral sequences that was held at Málaga, October 18--20 (2016), during the "Fifth Young Spanish Topologists Meeting". The approach was to illustrate the basic notions via fully computed examples arising from Algebraic Topology and Group Theory.

On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups

We compute the cotorsion product of the mod 2 cohomology of spinor group spin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of the classifying space of the spinor group spin(n). As a consequence of this computation, we show the non-collapsing of the Rothenberg-Steenrod spectral sequence for n > 16.

A generalization of the stable EHP spectral sequence

For any vector bundle, we define an inverse system of spectra. In the case of a trivial bundle over a point, the homotopy groups of the filtration quotients give rise to the stable EHP spectral sequence, as was shown by Mahowald. The limit was determined by Lin. We also obtain a spectral sequence and show strong convergence in certain special cases. For compact base spaces, we obtain a generalization of Lin's theorem. In the case of the universal bundle over BO, we can also determine the limit by means of an Adams spectral sequence. This turns out to be quite different from the compact case. W…

A-homology, A-homotopy and spectral sequences

Given a CW-complex A we define an `A-shaped' homology theory which behaves nicely towards A-homotopy groups allowing the generalization of many classical results. We also develop a relative version of the Federer spectral sequence for computing A-homotopy groups. As an application we derive a generalization of the Hopf-Whitney theorem.

The representation-ring-graded local cohomology spectral sequence for BPR<3>

The purpose of this paper is to examine the calculational consequences of the duality for BPR<n> proved by the first author and Meier in 1607.02332, making them explicit in the special case n=3. We give an explicit description of the behaviour of the spectral sequence, with pictures.