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A question about spectral sequences

In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
Mehmet Onat's user avatar
  • 1,211
4 votes
2 answers
236 views

Loop-space functor on cohomology

For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$. More concretely, $\omega$ is given by the Puppe sequence $$\...
Leo's user avatar
  • 633
1 vote
0 answers
72 views

A question about the localization theorem of Borel-Hsiang and spectral sequence

Suppose that $T$ is a torus acting on a topological space $X $. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
Mehmet Onat's user avatar
  • 1,211
0 votes
0 answers
38 views

Generalized edge map in spectral sequence of double complex

suppose we have a cohomologically indexed double complex $C^{\bullet,\bullet}$ with its spectral sequence $$E_2^{p,q}=H^p_{vert}(H^q_{horiz}(C))\Rightarrow H^{p+q}(C)$$ and suppose that the horizontal ...
xir's user avatar
  • 1,984
2 votes
2 answers
97 views

Extensions of $G$-modules parametrized by $H^1$

Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
Conjecture's user avatar
12 votes
5 answers
2k views

What are some good examples of spectral sequences which degenerate after the first nontrivial differential?

The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
19 votes
0 answers
301 views

Are these comparison morphisms between Čech and Grothendieck cohomology the same?

For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
FShrike's user avatar
  • 831
2 votes
0 answers
152 views

Vanishing differential of Brown-Gersten-Quillen spectral sequence

Let $k$ be an algebraically closed field of characteristic zero and $X$ be an affine, simplicial toric 3-fold over the field $k$. I am trying to use the Brown-Gersten-Quillen spectral sequence to ...
Boris's user avatar
  • 549
9 votes
1 answer
229 views

Is there a correction to the failure of geometric morphisms to preserve internal homs?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
Cameron's user avatar
  • 121
4 votes
1 answer
373 views

Two spectral sequences arising from a simplicial spectrum

Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective. From this situation, we can form two filtrations on $X$: the ...
Brian Shin's user avatar
2 votes
0 answers
160 views

Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory

$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2. ...
JeCl's user avatar
  • 1,001
2 votes
0 answers
96 views

Alternative construction for the loop space (?)

There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
Andrea Marino's user avatar
1 vote
1 answer
349 views

Why should we study the total complex?

Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
mrtaurho's user avatar
  • 165
0 votes
0 answers
105 views

Basis of Lambda algebra for a programmer

First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
Dmitry Vilensky's user avatar
2 votes
0 answers
125 views

Homotopy equivalence of chain complexes from subcomplexes and quotient complexes

Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
Faniel's user avatar
  • 653
10 votes
2 answers
755 views

Spectral sequences and short exact sequences

Suppose I take a short exact sequence of filtered chain complexes: $$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$ We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
Richard Hepworth's user avatar
1 vote
0 answers
100 views

Spectral sequence for a truncated semi cosimplicial space

Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
happymath's user avatar
  • 167
4 votes
1 answer
198 views

Collapse of spectral sequence computing Equivariant cohomology

I have already posted this question on math.stackexchanges but I got no answer and I decided to post it here. Let us consider the fibration $$ M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG ...
RiemannGauss's user avatar
5 votes
0 answers
469 views

Hypercohomology spectral sequence from the derived category point of view

Let $F\colon \mathsf{A}\to\mathsf{B}$ be an additive functor between abelian categories and let $M$ be a complex on $\mathsf{A}$. There's a "hypercohomology spectral sequence" $$E_1^{i,j}=\...
Gabriel's user avatar
  • 1,049
6 votes
1 answer
343 views

Different flavours of Vassiliev Conjecture

There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
Andrea Marino's user avatar
3 votes
1 answer
415 views

Spectral sequence in Adams's book, Theorem 8.2

I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
T. Wildwolf's user avatar
2 votes
0 answers
277 views

Notation for spectral sequences [closed]

Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth ...
Andrea Marino's user avatar
4 votes
2 answers
372 views

Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups

I have a very soft question which might be very standard in textbooks or literature but I haven't seen it. To a fixed group $G$ we may attach different topologies to make it different topological ...
XYC's user avatar
  • 389
1 vote
1 answer
136 views

Relation between $G_0(X)$ and $\mathrm{Cl}(X)$ for a normal variety $X$

Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$. I am trying to show that there is a surjective group homomorphism $G_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by ...
Boris's user avatar
  • 549
1 vote
0 answers
84 views

Image of the boundary maps in the homological spectral sequence of a filtration of a chain complex

I'm trying to understand the construction of the homological spectral sequence of a filtration given in C.A.Weibel ''An introduction to homological algebra''. Here, they start with a filtration of a ...
Marcos's user avatar
  • 577
16 votes
1 answer
762 views

The second stable homotopy group

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
Leo's user avatar
  • 633
0 votes
1 answer
219 views

Infinite dimensional homology and spectral sequences

I am new to spectral sequences, so I'm not sure about the difficulty of this question. Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and ...
Marcos's user avatar
  • 577
6 votes
1 answer
446 views

Leray spectral sequence and pullbacks

I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence: Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
Victor de Vries's user avatar
2 votes
1 answer
229 views

For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?

Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows? $$...
Suzet's user avatar
  • 697
0 votes
0 answers
185 views

Interpreting the edges in the Serre spectral sequence

Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
SpectralSequenceGuy's user avatar
1 vote
0 answers
164 views

Spectral sequence for two fibrations

Given maps of fibrations, i.e. commutative diagrams of smooth manifolds $$\begin{matrix} \ F & \to & E &\to & B \\\ \downarrow & & \downarrow & & \downarrow \\\ \ F'...
UserIn's user avatar
  • 103
4 votes
0 answers
170 views

infinite families in stable homotopy groups

The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic. But I wonder if the order of Mahowald's elements is known? in Green Book it mentioned ...
Dr.Martens's user avatar
4 votes
0 answers
97 views

"Standard computations" with stable Hopf invariants

I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...
Tommaso Rossi's user avatar
7 votes
1 answer
237 views

Relation between cohomology operations and the Adams spectral sequence

$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Cone}{Cone}$ I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
Shivang's user avatar
  • 71
2 votes
1 answer
203 views

Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering $$ \dotsb \...
KKD's user avatar
  • 463
3 votes
0 answers
220 views

Explicit description of the Leray spectral sequence with compact supports for a fibration

Consider a locally trivial fibration $f: E \to B$ with fiber $F = \mathbb{C}^n$. The Leray spectral sequence with compact supports is $$ E_2: H^p_c(B, \underline{H^q_c(F)}) \implies H^{p+q}_c(E). $$ ...
Eduardo de Lorenzo's user avatar
3 votes
0 answers
224 views

Explicit computation of hyper Ext in terms of the homologies of the input chain complexes

This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello! Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
Eric's user avatar
  • 301
7 votes
1 answer
412 views

Basic example of derived descent

I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example. Given a ...
Jake McNamara's user avatar
5 votes
1 answer
355 views

Understand the proof that rational resolution is independent of the resolution

EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...
Invariance's user avatar
2 votes
0 answers
93 views

Name for the "other term" in a derived exact couple

I'm building a spectral sequence using an exact couple $D^1 \to D^1 \to E^1 \to D^1$, with $k$th derived exact couple $D^k \to D^k \to E^k \to D^k$. In this case, I happen to have more information ...
Colin Aitken's user avatar
2 votes
0 answers
201 views

Grothendieck spectral sequence and exact couples

I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges ...
user avatar
2 votes
0 answers
174 views

When does Tate spectral sequence degenerate at $E_2$?

For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence $$ E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
user145752's user avatar
4 votes
1 answer
296 views

Higher order differentials of Bockstein spectral sequence

The Bockstein SS is obtained from the exact sequence $$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$ with $E_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d_1=Sq^1$. How to identify ...
Nanjun Yang's user avatar
3 votes
0 answers
67 views

Tensor product of exact couples

Suppose $(D_1,E_1;i_1,j_1,k_1)$ and $(D_2,E_2;i_2,j_2,k_2)$ are two exact couples, ie., there are exact sequences $D_1\xrightarrow{i_1}D_1\xrightarrow{j_1}E_1\xrightarrow{k_1}D_1$ and $D_2\xrightarrow{...
Faniel's user avatar
  • 653
3 votes
0 answers
117 views

The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$

$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...
Leo's user avatar
  • 633
1 vote
0 answers
85 views

Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence

Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
The Thin Whistler's user avatar
4 votes
1 answer
168 views

$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$

What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$-ANSS for $S/3$?
Jordan Levin's user avatar
4 votes
1 answer
378 views

The Hochschild–Serre spectral sequence and cup products

Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences. \begin{align*} E_2^{pq}: H^p(k, H^q(X_{...
Tim Santens's user avatar
2 votes
0 answers
125 views

Frölicher spectral sequence of a surface

Asked this on MSE but didn't get much attention. Let $ S $ be a compact complex surface. Can anyone provide a proof of the fact that the Frölicher spectral sequence of $ S $ degenerates at $ E_1 $? ...
Cranium Clamp's user avatar
3 votes
0 answers
162 views

Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?

Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
Andrea Marino's user avatar

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