# 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 & Tu - Spectral Sequences

I have been reading this book with the main purpose of introducing myself to spectral sequences. Although so far I've been enjoying it quite a lot, I've had a problem I cannot handle: In theorem 14.6 ...
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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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### A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor $E_{r-1}$...
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### Extension problem for serre spectral sequence of trivial fibration

Consider the cohomology spectral sequence of the trivial fibration $F \hookrightarrow F \times B \to B$. I showed in my answer here Why all differentials are $0$ for Serre Spectral Sequence of ...
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### Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
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### basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
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### Cohomological AHSS for projective space $\mathbb{C}P^n$ ALWAYS collapses at the second page

While I was computing the cohomology ring $E^*(\mathbb{C}P^n)$ for $E$ an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ...
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### When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
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### unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...