# 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.

265 questions
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### Can there be nonisomorphic functorial Serre spectral sequences?

The Serre spectral sequence is a very useful tool in algebraic topology, but as often with these beasts, the differentials can be hard to compute. Of course, the proof that this sequence exists is ...
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### Chapter 1 in A User's Guide to Spectral Sequences, reconstruction

In Chapter 1 of John McCleary's book A User's Guide to Spectral Sequences the condition assumed for the filtrations is that they are bounded below. In pg. 4 it is said that $H^*$ (if good enough) can ...
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### Serre classes and the Serre spectral sequence

Let $C$ be a Serre class which satisfies the additional axioms about $\otimes, \mathrm{Tor}, K(A,1)$'s. It is then easy to check that if $F\to X\to B$ is a Serre fibration with $\pi_1(B)$ acting ...
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### Convention for the grading in spectral sequences

Usually spectral sequences are defined together with a grading on it, the differential $d_r$ on the page $r$ taking $E_r^{p,q}$ to $E_r^{p-r, q+r-1}$. This convention on the grading on the ...
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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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### Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
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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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### Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
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### Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
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### Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
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### What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
### What does it mean that if the cover of $k$-fold intersections is not contractible it takes the form of a spectral sequence of cohomology?
I asked the following question in a previous post: Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only ...
Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$. The proof I ...