Comultiplication in the Serre Spectral Sequence

动机构上同调的Serre型谱序列

A Serre-type spectral sequence for motivic cohomology

本文构造了动机构上同调的Serre型谱序列，并将其应用于射影一般线性群Nisnevich分类空间的动机构上同调计算，从而给出Severi-Brauer簇动机的显式描述。

Fibered Symplectic Cohomology and the Leray-Serre Spectral Sequence

We define Symplectic cohomology groups for a class of symplectic fibrations with closed symplectic base and convex at infinity fiber. The crucial geometric assumption on the fibration is a negativity property reminiscent of negative curvature in complex vector bundles. When the base is symplectically aspherical we construct a spectral sequence of Leray-Serre type converging to the Symplectic cohomology groups of the total space, and we use it to prove new cases of the Weinstein conjecture.

Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Kh…

Eilenberg-Moore和Leray-Serre谱序列的前序

Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences

本文探讨了具有相同极限的Leray-Serre和Eilenberg-Moore谱序列的关系，提出联合三分次细化及两个前序谱序列，揭示其一恒退化、另一具局部到全局退化性质。

On Sikora's spectral sequences

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.

Spectral Sequences For Commutative Lie Algebras

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.

热带曲线模空间的Serre谱序列

A Serre spectral sequence for the moduli space of tropical curves

该研究构造了适用于热带曲线模空间的Serre谱序列，可通过图上点构形空间的紧支上同调计算其等变约化有理上同调，并借助等变同构得到特定亏格与点数范围的模空间权0紧支有理上同调结果。

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…

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.

String spectral sequence

We define shriek map for a finite codimensionnal embedding of fibration. We study the morphisms induced by shriek maps in the Leray-Serre spectral sequence. As a byproduct, we get two multiplicative spectral sequences of algebra wich converge to the Chas and Sullivan algebra $\mathbb{H}_*(LE)$ of the total space $E$ of a fibration. We apply this technic to find some result on the intersection morphism $I: \mathbb{H}_*(LE) \longrightarrow H_*(ΩE)$ and to the space of free paths on a manifold $M^I$.

Derived string topology and the Eilenberg-Moore spectral sequence

Let $M$ be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop…

On a spectral sequence for the cohomology of infinite loop spaces

We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a connective spectrum $X$. Under good conditions its $E_{2}$-term is computable as certain non-abelian derived functors evaluated at $H^*(X)$ as a module over the Steenrod algebra, and it converges to the cohomology of $Ω^\infty X$. We provide general methods for computing the $E_{2}$-term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples a…

交错Mayer-Vietoris谱序列

Interleaving Mayer-Vietoris spectral sequences

本文将Mayer-Vietoris谱序列引入持久同调作为不变量，提出$\varepsilon$-无环承载子等概念并研究其稳定性，最终证明了近似神经定理与谱序列的$\varepsilon$-交错条件。

Computing a spectral sequence of finite Heisenberg groups of prime power order

Let $p\geq 5$ be a prime number, let $n\geq 2$ be a natural number and let $\text{Heis}(p^n)$ denote the Heisenberg group modulo $p^n$. We study the Lyndon-Hochschild-Serre spectral sequence $E(\text{Heis}(p^n))$ associated to $\text{Heis}(p^n)$ considered as a split extension, and show that, $E(\text{Heis}(p^n))$ collapses in the third page. Moreover, for a fixed $p$, the spectral sequences $E(\text{Heis}(p^n))$ are isomorphic from the second page on.

A spectral sequence for fusion systems

We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove an analogue of a result of Stallings in the context of fusion preserving homomorphisms and deduce Tate's p-nilpotency criterion as a corollary.

Discrete Vector Fields and Fundamental Algebraic Topology

We show in this text how the most important homology equivalences of fundamental Algebraic Topology can be obtained as reductions associated to discrete vector fields. Mainly the homology equivalences whose existence -- most often non-constructive -- is proved by the main spectral sequences, the Serre and Eilenberg-Moore spectral sequences. On the contrary, the constructive existence is here systematically looked for and obtained.

关于超实数双斯蒂罗代数

On the hyperreal dual Steenrod algebra

本文计算了在特定系数下的双斯蒂罗代数，并利用同调谱序列研究了相关同调结构。

Successive Spectral Sequences

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…

Characteristic classes for cohomology of split Hopf algebra extensions

We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.

The homotopy Leray spectral sequence

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.

Equivariant local coefficients and the RO(G)-graded cohomology of classifying spaces

This thesis consists of two main parts. In the second part, 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 for RO(G)-graded Bredon homology and cohomology. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology o…

On Leibniz cohomology

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with adjoint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of (the dual of) a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.

Spectral Sequences in String Topology

In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundles: the fibre bundle $Ω^n M \to L^n M \to M$ for an $h_*$-oriented manifold $M$ and the looped fibre bundle $L^n F \to L^n E \to L^n B$ of a fibre bunde $F \to E \to B$ of $h_*$-oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We a…

紧李群等维作用的谱序列

A Spectral Sequence for Equidimensional Actions of Compact Lie Groups

本文为紧连通李群在紧流形上的等维作用建立了Leray-Serre谱序列，刻画其第二页并联系轨道空间、李代数与流形的上同调，且通过爆破方法推广至非等维情形，给出流形上李群作用不存在的拓扑障碍。

Cohomology of Split Group Extensions and Characteristic Classes

There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.

代数上作用素的切余结构的谱序列

A spectral sequence for tangent cohomology of algebras over algebraic operads

本文提出了一种用于计算代数上操作符切余同调的谱序列，该序列可应用于拓扑空间的有理Adams-Hilton构造，并揭示了Serre谱序列的代数描述及其收敛性。

Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this program, we establish the properties of stratified fibrations over unfiltered base spaces and of their mapping cylinders. We al…

A Division Theorem for Nodal Projective Hypersurfaces

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…

历史脉络中的齐性空间上同调

The cohomology of homogeneous spaces in historical context

本文从历史角度梳理齐性空间上同调的计算发展，说明其如何推动极小模型、Eilenberg-Moore 谱序列和 A-无穷代数等工具的诞生。