# 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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### Collapsing of Kunneth formula for equivariant K-theory of homogeneous spaces.

Minami in "K-groups of symmetric spaces" (equations 1.1, 1.2) states the following, originally due to Hodgkins: Suppose that $G$ is a compact connected Lie group such that $\pi_1(G)$ is ...
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### Computing a Massey product.

Here is the question I am trying to solve: Can anyone help me in showing me how to compute this Massey Product?
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### Do we need $M$ simply connected for $H^*(E)=H^*(M)\otimes H^*(F)$ for $E$ a fibre bundle?

I'm following notations from sec. 14 Bott and Tu here: Suppose $\pi :E\rightarrow M$ be a fibre bundle on a manifold with fibre $F$ and $\mathcal{U}=\{U_\alpha\}_\alpha$ be a good cover of $M$. Then ...
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### Homology of the diagonal sequence of 3x3 commutative diagram of modules

Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules), and suppose that we have a 3x3 commutative diagram, where ...
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### What is "spectral development" of a linear operator?

I want to find the "spectral development" (or spectral series, spectral decomposition, not sure how to translate to english) of the linear operator. What is that? I can't even google it. I ...
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### Some confuse in spectral sequence and its calculate

We have the Leray's theorem: Let $\pi:E\longrightarrow B$ be a fiber bundle with fiber $F$ over a simply connected base space $B$. Assume that in every dimension $n$, $H^{\ast}(F)$ is of finite rank ...
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