# 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.

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### Is there a spectral sequence for Ext associated to an $n$-term exact sequence of modules?

I ask this question because of the following fact: Let $R$ be a ring and $0 \to M_1 \to M_2 \to M_3 \xrightarrow{\alpha} M_4 \to 0$ be an exact sequence of $R$-modules. Suppose we know $Ext_R(M_1,R)$,...
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### Isomorphic total complexes, spectral sequence argument

I would like to proof the following claim: Let $f: C_{\ast,\ast}\to C'_{\ast,\ast}$ be a map of bicomplexes (differentials anticommute) that is a quasi-isomorphism restricted to each column. Then $f$...
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### Computing oriented Bordism groups of a manifold $X$ for $n\leq 4$

I'm following the book of Davis and Kirk "Lecture notes in Algebraic Topology" where, at pages $246-247$ there is a computation of the aforementioned groups via the Atiyah- Hirzebruch Spectral ...
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### 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.
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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,...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### $h^{0, 1}$ 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 ...
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### 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 ...