# 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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### Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
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### Product structure in Cohomology Spectral sequence

In the Serre Spectral sequence, we know, the cup product structure induces a canonical product in all $E_{r}$ pages which is compatible with respect to the differential. I am trying to understand ...
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### Group actions on spectral sequences of group cohomology

Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
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### Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.

It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
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### Thom spectrum $MSpin$ and $E_2$-page for a large degree $i\geq 8$

Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$. For a computation involving no odd torsion, the Adams ...
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### functoriality of Grothendieck spectral sequence

I am looking for a reference which treats the functoriality of the Grothendieck spectral sequence for elements of the derived category of an abelian category.
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### Cohomolgoy of $S^1$-bundle over genus $g$ surface

Suppose $\Sigma_g$ is the closed, orientable surface of genus $g\ge 1$. Isomorphism classes of principal $S^1$-bundle on $\Sigma_g$ is then classified by $H^2(\Sigma_g,\mathbb{Z})=\mathbb{Z}$. Suppose ...
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### First page of a spectral sequence

I have a very basic problem when trying to understand "You could have invented spectral sequences" by Timothy Chow (but I will index cohomologically since I'm more interested in cohomology). This is ...
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### Leray spectral sequence

Let $f : X\to Y$ be a continuous map of topological spaces, $A$ an abelian sheaf on $X$. We have the Leray spectral sequence $$E_2^{p,q} := H^p(Y, R^qf_*A)\Rightarrow H^{p+q}(X, A).$$ Could someone ...
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In this article by Snaith (p. 575) appears the following comment: ... these transgressive elements [...] can be located by means of the Adams operations [...]. These operate (unstably) in both the ...
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### Cohomology of augmented double complex with exact rows using spectral sequences

I am learning about spectral sequences through Vakil's notes and wanted to try out a simple spectral sequence argument on a double complex. I recalled this little paragraph in Bott & Tu, p.97: ...
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### Homology with coefficients from homology.

My main goal is to understand the computations behind the cohomology ring of $\mathbb{C}P^n$ as done in Bott & Tu. To this ends, I am reading a set of notes about Spectral Sequences (here) by ...
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### Hochschild-Serre spectral sequence

I have a hopefully simple question about the Hochschild-Serre spectral sequence (which may just be a simple question about general spectral sequences). Whenever I see the sequence written down, it has ...
Let $(C^{p,q},d_h,d_v)$ be a double complex of modules and let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that (1) the horizontal rows are exact, (2) the columns ...