# 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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### Bott and Tu Proposition 12.1

Question about the proof of Bott and Tu's Proposition 12.1: Given any double complex $K$, if $H_\delta H_d(K)$ has entries only in one row, then $H_\delta H_d$ is isomorphic to $H_D$. Remark: $\delta$...
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### Bott and Tu Spectral Sequence of a Double Complex

This page has confused me a great deal, I have a few questions: Is $A^k$ the same thing as $C^k = \oplus_{p + q = n} K^{p,q}$ as defined in example 14.2 or is it different because $A = \oplus K_p$ ...
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### Does the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence go under a different name?

This is more of a naming convention question than anything else. I am reading up on spectral sequences mainly from the book Lecture Notes in Algebraic Topology by Davis and Kirk. Now in this book the ...
1answer
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### Logarithmic de Rham complex on a curve and the Hodge spectral sequence

Given a smooth projective curve $C$ over a field $k$ of characteristic $0$, let $D$ be a divisor. Let $\Omega_C^{\bullet}(D)$ be the logarithmic de Rham complex of $C$, with log poles along $D$. Is ...
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### Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$. In the beginning he states that there exisits ...
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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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### 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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### Homotopy groups $\pi_{n+1}(S^{n})$

In section $5.1$ of Hatcher's note about spectral sequences, he starts to compute stable homotopy group $\pi_{n+k}(S^{n}),k \leq 3$. Particularly for $\pi_{n+1}(S^{n})$, by Freudenthal suspension ...
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### Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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### degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
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### Spectral sequences on their own right

I have begun to read a bit about Spectral Sequences. I have been studying some topics of algebraic topology and this "tool" came up from time to time. I didn't need to go deep, but I felt I had to ...