# 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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### Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
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### Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
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### Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
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### What can we say about the category of spectral sequences?

For simplify, let's just consider the following category first: object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian ...
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### 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 ...
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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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### Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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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 ...
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### Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
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### Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
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### What's wrong with this argument that the Atiyah-Hirzebruch spectral sequence always degenerates?

Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form: $H^s(X,E^t) \Rightarrow E^{s+t}(X)$ This is a half-...