David E. Hurtubise
In this note we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author's work on Morse-Bott homology with A. Banyaga.
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The aim of this paper is to give a thorough insight into the relationship between the Rumin complex on Carnot groups and the spectral sequence obtained from the filtration on forms by homogeneous weights that computes the de Rham cohomology of the underlying group.
We present a family of model structures on the category of multicomplexes. There is a cofibrantly generated model structure in which the weak equivalences are the morphisms inducing an isomorphism at a fixed stage of an associated spectral sequence. Corresponding model structures are given for truncated versions of multicomplexes, interpolating between bicomplexes and multicomplexes. For a fixed stage of the spectral sequence, the model structures on all these categories are shown to be Quillen equivalent.
Spectral complexes from truncated multicomplexes
本文提出了一种基于截断多复形的谱复形构造方法，用于研究子黎曼几何，特别是Carnot群的结构。
Model category structures on truncated multicomplexes for complex geometry
本文在 N-多重复形上构造了一族模型范畴结构，其弱等价由两个谱序列的固定页上的拟同构刻画，从而为研究近复与广义复流形的同伦型提供基础。
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.
We construct a spectral sequence for computing KR-theory, analogous to the spectral sequence relating motivic cohomology to algebraic K-theory.
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
Let X be a finite, n-dimensional, r-connected CW complex. We prove the following theorem: If p \geq n/r is an odd prime, then the loop space homology Bockstein spectral sequence modulo p is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.
We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group.
We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah-Hirzebruch type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones to compute algebraic K- and L-groups of group rings and topological K-groups of reduced group C*-algebras.
A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence constructed by Cohen, Jones and Yan for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-p cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-p chains on its loop-space carries a theory of support varieties.
A motivic Greenlees spectral sequence towards motivic Hochschild homology
本文定义动机Greenlees谱序列，研究动机上同调谱模素数p的动机拓扑Hochschild同调，并在代数闭域特征与p互素时确定相关谱的同伦环。
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.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion.
In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.
It is shown that the Grayson tower for $K$-theory of smooth algebraic varieties is isomorphic to the slice tower of $S^1$-spectra. We also extend the Grayson tower to bispectra and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$-theory spectrum $KGL$. This solves Suslin's problem for these two spectral sequences in the affirmative.
This is a survey of author's results on weight structures and Voevodsky's motives. Weight structures are natural counterparts of t-structures (for triangulated categories) introduced by the author. They allow to construct weight complexes, weight filtrations, and weight spectral sequences for various cohomology theories. Partial cases of the latter are: 'classical' weight spectral sequences (for singular and etale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. All of those are mentioned in the current paper. The details, proofs, and several more results cou…
We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories including Hochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen and others.
The paper summarizes the construction of pairings on some standard spectral sequences in algebraic topology.
We show the homological Serre spectral sequence with coefficients in a field is a spectral sequence of coalgebras. We also identify the comultiplication on the $E^2$ page of the spectral sequence as being induced by the usual comultiplication in homology. At the end, we provide some example computations highlighting the use the co-Leibniz rule.
We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain comparison spectral sequences which mediate between Chevalley-Eilenberg-, commutative-and Leibniz cohomology. These methods are illustrated by a few computations.
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…
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.
In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphims between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to oth…
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。
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…
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 $\mathbb{C}$.
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.
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.
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。
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…
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 Swan, Leray and Leray-Serre spectral sequences of a compact Hausdorff HLC space acted on by a discrete group are shown to be isomorphic.
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.
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 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.
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…
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: 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。
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.
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.
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.
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.
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).
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 purpose of this short article is to record the computation of the homotopy groups of 3-local $\mathrm{tmf}$ via the Adams 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].
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.
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。
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.
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.
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。
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.
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.
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.
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.
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).
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…
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-…
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.
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