# 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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### 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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### Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy answer,...
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### Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
118 views

### Calculating the extraordinary cohomology of $\mathbb{C}P^n$

Let $E$ be a ring spectrum with an orientation. Now I want to calculate $E^*(\mathbb{C}P^n)$. The definition of orientation I am using is: There is an element $x \in E^*(\mathbb{C}P^{\infty})$ such ...
988 views

### Hopefully an easy question on spectral sequences

I'm trying to understand Proposition 4.3 (page 562) in S. Morita's article Characteristic Classes of Surface Bundles, which can be found on Andy Putman's website here. I don't think that my question ...
350 views

### Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
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### cohomology ring structure of conf($\mathbb{R}^m$, 3)

I am attempting to compute the (integral) cohomology ring structure of the 3 configuration space of $\mathbb{R}^m$ and have run into a few doubts. Using a result of Fadell and Neuwirth, we have that ...
304 views

### Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
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### What is a spectral sequence?

Can anyone explain what a spectral sequence is? What is the motivation behind this? Is it a natural tool? Why should we study spectral sequences? Pardon me for asking too many questions. Actually I ...