# 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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### Serre Spectral sequence [Hatcher]

I'm having trouble understanding Serre Spectral sequence given in Hatcher. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf In the beginning of the section, (Page 526 of the book, or rather, Pg 9 of ...
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### Differentials on the second page of the spectral sequence of a first quadrant double complex

Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to ...
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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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### Spectral sequence $B^1_{p,q}= B^\infty_{p,q}$ definition, nlab

In an introductory notes to spectral sequences, nlab, Definition 1.26 we define $$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$ and $$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})$$ So what is the ...
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### Spectral Sequence and Stiefel Manifold

Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration: $$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$ Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
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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-...
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### Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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### Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
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### Why does spectral sequence $E^\infty$ need AB4*

So in Weibel, he states Warning: In an unbounded spectral sequence, we will tacitly assume that $B^{\infty}$, $Z^{\infty}$, and $E^{\infty}$ exist! The reader who is willing to only work in the ...
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