# 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.

17 questions
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### Bockstein homomorphism and Steenrod square

question: What is the relation between Bockstein homomorphism and Steenrod square? For example, can one explain why the following relation works in the case of cohomology group with $\mathbb{Z}_2$ ...
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### How to show that $0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0$ is exact?

Suppose that a spectral sequence converging to $H_\ast$ has $E_{pq}^r = 0$ for all $p\neq 0,1$. Show that there are exact sequences $$0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0 \,.$$ ...
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### Another way to compute $\pi_4(S_3)$: contradiction in spectral sequence calculation

$\newcommand{\Z}{\mathbb{Z}}$ I decided that I would try another way of computing $\pi_4(S_3)$. Take the fibration $S_3 \to K(\Z,3)$ with fiber defined to be $X_4$. I want to directly use this ...
I want to start by saying that I only have very basic notions about spectral sequences. Consider a short exact sequence of chain complexes 0\longrightarrow A\longrightarrow B\longrightarrow C\...