Questions tagged [spectral-sequences]

Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.

Filter by
Sorted by
Tagged with
1
vote
0answers
146 views

Bott & Tu - Spectral Sequences

I have been reading this book with the main purpose of introducing myself to spectral sequences. Although so far I've been enjoying it quite a lot, I've had a problem I cannot handle: In theorem 14.6 ...
4
votes
0answers
113 views

Edge homomorphisms for Atiyah-Hirzebruch spectral sequence for a spectrum $X$

I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$. $$\times \times \times $$ Let $X$ be a connective spectrum, as ...
7
votes
1answer
249 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor $E_{r-1}$...
5
votes
0answers
105 views

A problem with a definition of homology of a spectrum

I'm currently reading some notes about the James spectral sequence (here) and there is a passage which is bothering me (page 749 bottom): $$ colim_n h_{p+q}(T(\xi_n))\cong h_{p+q}(M\xi)$$ where $M\...
3
votes
1answer
97 views

Reference Request: The Atiyah-Hirzebruch Spectral Sequence

I have just finished learning the Serre spectral sequence and I would like to learn about the Atiyah-Hirzebruch spectral sequence. Could someone suggest an accessible reference? Thank you in advance.
1
vote
0answers
73 views

Edge morphisms coincide cup-products in the Tate spectral sequence

In the Tate spectral sequence, the edge morphism coincides with the cup product. The proof is written in Neukirch-Schmidt-Wingberg's book: Cohomology of Number Fields (Theorem 2.5.5,p125). https://www....
2
votes
0answers
22 views

What is the action on cycles

Let $A$ be a graded decreasing filtered chain complex, with cohomology differential $d$. Let $^nA^p=A^{p,n-p}$ be the elements of $A$ with total degree $n$, filter degree $p$, (and therefore with ...
6
votes
2answers
538 views

What is a spectral sequence?

Can anyone explain what a spectral sequence is? What is the motivation behind this? Is it a natural tool? Why should we study spectral sequences? Pardon me for asking too many questions. Actually I ...
3
votes
0answers
83 views

Calculating torsion in $\pi_i(S^{2n})$

Let $p$ be an odd prime and $n>1$. I want to prove that $\pi_i(S^n)$ has no $p$-torsion for $i<n+2p-3$. For odd $n$ this is Proposition 6.26 in McCleary's book (p.206). He mentions afterward ...
1
vote
1answer
58 views

H-spaces act trivially on fiber?

If $F\to E\to B$ is a fibration and $B$ is a path-connected H-space, is the action of $\pi_1(B)$ on $H^*(F)$ by fiber transport always trivial? The reason I am asking is that I would like to consider ...
1
vote
1answer
71 views

$E_{p, 0}^2$ and $E_{0, 1}^2$ terms in sequence, in terms of homology of $K(G, 1)$, homology of $K(R, 1)$, and action of $G$ on $R$ by conjugation?

This is a followup to my previous question, reproduced here. Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, ...
4
votes
1answer
244 views

Kernel of Hurewicz map using the spectral sequence of the universal cover

In Davis & Kirk's Lecture Notes in Algebraic Topology, Exercise 159 reads: Use the spectral sequence of the universal cover to show that for a path-connected space $X$ the sequence $$\pi_2(X)\...
4
votes
2answers
275 views

Surjectivity of Edge morphism in A-H cohomology Spectral Sequence

As the title suggests, I'm interested in proving the following claim: Recall the AH-spectral sequence:$$ E_2^{pq}=H^p(X,\mathcal{H}^q(\ast)) \Longrightarrow \mathcal{H}^{p+q}(X)$$ and since $\...
2
votes
0answers
96 views

Extension problem for serre spectral sequence of trivial fibration

Consider the cohomology spectral sequence of the trivial fibration $F \hookrightarrow F \times B \to B$. I showed in my answer here Why all differentials are $0$ for Serre Spectral Sequence of ...
4
votes
1answer
187 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
6
votes
1answer
343 views

Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
2
votes
1answer
120 views

Exact sequence from Serre spectral sequence

let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid.. It regards the Serre spectral sequence associated to a fibration $0\rightarrow F ...
0
votes
1answer
45 views

Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
1
vote
0answers
36 views

basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
1
vote
0answers
33 views

Analogue of spectral sequences for simplicial sets

Is there an analogue to spectral sequences where, instead of chain (bi)complexes, we use simplicial sets? Namely, let $\{\mathbf{S}_{p,q}\}_{p,q}$ be sets such that $\mathbf{S}_{p,\bullet}$ and $\...
0
votes
1answer
169 views

Partial Converse to “Pushout of a cofibration is a cofibration”

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I want a converse of this fact specialized to the case where I am pushing out a map BY a fibration: I.e., if I am given a diagram $ \begin{CD} E_1 @&...
1
vote
0answers
166 views

When does the Grothendieck spectral sequence converge?

I am trying to understand spectral sequences in algebraic geometry. One has the Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$, and $\mathcal G: \...
1
vote
1answer
45 views

Why is $F_{n,0}=H_n(K)$ for an arbitrary filtered complex?

Let $... \subset K_{-1}=0 \subset K_0\subset ...K_n \subset...$ be an arbitrary filtered chain complex with $colim_n K_n:=K$. Let $F_{p,p+q}=im(H_{p+q}(K_p) \to H_{p+q}(K))$ Mosher and Tangora ...
6
votes
1answer
90 views

Computing the “limit” of a SSeq with $E_{\infty}^{*,*}$ a free graded $\Gamma$-module.

I'm trying to prove the following proposition from Kochman's book. For completion I will write it here the relevant part: Let $E$ be an oriented spectrum with orientation class $x\in E^2(\mathbb{C}...
2
votes
1answer
69 views

Cohomological AHSS for projective space $\mathbb{C}P^n$ ALWAYS collapses at the second page

While I was computing the cohomology ring $E^*(\mathbb{C}P^n)$ for $E$ an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ...
2
votes
0answers
32 views

Pairing on the AHSS induced by cap product: why does it exists

This is my setting: Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)\...
1
vote
0answers
30 views

Rational homology of $\Omega^{n+1}\Sigma^{n+1}X$

I want to know how compute, by induction and using the Serre spectral sequence for homology, $H_*(\Omega^{n+1}\Sigma^{n+1}X, \mathbb{Q})$. I know that I have to use the path-loop fibration $$ \Omega^...
4
votes
0answers
91 views

When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
1
vote
0answers
20 views

unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...
1
vote
2answers
64 views

Contradiction in spectral sequence calculation of $H_*(BO(2))$

$\newcommand{\Z}{\mathbb{Z}}$ For this post I am going to assume the answer namely $H_*(BO(2))=\Z_2[w_1,w_2]$. Consider the fibration $S^1 \hookrightarrow BO(1) \to BO(2)$. The $E^2$ page has $E^2_{i,...
3
votes
0answers
64 views

computing $\pi_1 S^1$ from a spectral sequence

In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a $1-sphere$. From this we are ...
1
vote
0answers
163 views

Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it's just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
3
votes
0answers
75 views

Comparison Criterion for Atiyah - Hirzebruch Spectral Sequence

Let us denote with $E(X)$ the A-H spectral sequence associated to a CW complex $X$ and homology theory $h_*$: $$ E(X)_{pq}^2 = H_p(X, h_q(\ast))\Longrightarrow h_{p+q}(X)$$ and with $E(Y)$ the one ...
2
votes
1answer
139 views

Generalized cohomology groups of torus

Let $\tilde h^\bullet$ be a reduced generalized cohomology theory, and let $T^2$ be the torus. For what theories $\tilde h^\bullet$ is $\tilde h^\bullet(T^2)$ known (or easily computable)? For ...
4
votes
1answer
492 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is connected,...
5
votes
1answer
268 views

Spectral sequences from Cartan-Eilenberg systems

This is an exercise from Mcleary's book on Spectral sequence which I have been stuck with for some time. Let us recall what a Cartan-Eilenberg system is: IT consists of a module $H(p,q)$ for each pair ...
5
votes
1answer
2k views

Long exact sequence for a triple follows from long exact sequence for a pair?

In homology theory, the long exact sequence for a pair $(X,Y)$ is just $H(Y)\to H(X)\xrightarrow{\partial(X,Y)}H(X,Y)\to H(Y)[-1]$. The long exact sequence for a triple $(X,Y,Z)$ is $H(Y,Z)\to H(X,Z)\...
4
votes
0answers
351 views

Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex $RF(...
1
vote
1answer
87 views

Definition of $E^{\infty}_{pq}$ terms in a spectral sequence. Something strange seems to happen

I'm trying to prove the following assertion in Weibel's Homological Algebra page 125, 5.2.8 Given a homology spectral sequence, we see that each $E^{r+1}_{pq}$ is a subquotient of the previous term ...
1
vote
1answer
104 views

Relation between long exact sequence and associated graded

I'm reading these notes by Hutchings on spectral sequences. In the first section, he motivates spectral sequences with the long exact sequence in relative homology. Given a chain complex $C_*$ and a ...
2
votes
0answers
126 views

Reference on Weibel's Homological Algebra: “$G/H$ acts by conjugation in LHS-spectral sequence”

I'm studying the Lyndon-Hochschild-Serre spectral sequence for $H\triangleleft G$: $$ H_p(G/H;H_q(H;A))\Rightarrow H_{p+q}(G;A) $$ where $A$ is a $G$-module. I was told (w/o giving a proof) that ...
3
votes
0answers
146 views

Mapping Lemma for Spectral Sequences in Weibel's book - Help with the proof

We start by recalling the definition of a morphism in the category of Spectral Sequences: a morphism $f \colon A \to E$ is a family of maps $f^r_{pq}\colon A^r_{pq}\to E^r_{pq}$ in the abelian ...
0
votes
1answer
198 views

Generating morphisms of spectral sequences

When we define spectral sequnces (as Weibel's book) for example in the abelian category $R$-mod, they are a collection of objects $E_{pq}^r$ for $p,q$ and $r\geq a$ integers with a collection of ...
5
votes
1answer
655 views

Total complex homology exact sequence

I'm been trying to do this problem (Problem 5.1.1) from Weibel's Introduction to Homological Algebra but I can't really see how to finish it. The statement of the problem is summarized as follows: ...
4
votes
0answers
141 views

$h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
3
votes
0answers
204 views

Basic questions about convergence of spectral sequences.

I have a fairly rudimentary understanding of spectral sequences. I have a couple questions though. In Algebra, Lang states what seems to me, to be two slightly different notion of convergence of a ...
1
vote
0answers
196 views

Grothendieck spectral sequence from the hypercohomology spectral sequence

Is it possible to write a proof of the convergence of the Grothendieck spectral sequence of the composition of two functors only using the convergence of the hypercohomology spectral sequences ...
1
vote
0answers
71 views

Leray sheaf being constant - what does it mean in terms of singular cohomology?

Let me first admit that I know next to nothing about sheaf cohomology, but I might have encountered a good reason to learn it. Suppose that I have a fibration $F \to E \to B$ and I know that its ...
2
votes
1answer
70 views

Using the Bockstein spectral sequence to identify direct summands

I have a question about demonstrating part 2 of corollary 5.9.12 in Weibel's An Introduction to Homological Algebra. Here is the setup. Fix a prime $p$ and suppose I have a long exact sequence of ...
2
votes
0answers
58 views

Spectral Sequence associated to a filtration abuts because we can find closed representatives

Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be ...