# 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.

264 questions
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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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### 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 ...
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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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### $h^{0, 1}$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy answer,...
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### How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to link/...
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### Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
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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 sequence page isomorphism

Suppose we have a map of spectral sequences $\{E_{p,q}^r,d^r\}\to \{{E'}_{p,q}^r,d'^r\}$, both generated from total chain complexes, $C$ and $C'$ respectively, such that for some $r$ the map between ...
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### A form of Künneth formula?

Problem (Exercise 15.12 in Bott & Tu's Differential Forms in Algebraic Topology) If $X$ is a space having a good cover, e.g., a triangularizable space, and $Y$ is any topological space, prove ...
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 $E_{r-1}$...