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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.

14
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1answer
636 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n &...
1
vote
1answer
440 views

Short exact sequence and cohomology group

Given a short exact sequence for some finite groups $A,B,C$, $$1\to A \to B \to C \to 1,$$ how could we construct an exact sequence of their cohomology group out of it? One version of the story I ...
8
votes
1answer
908 views

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ ...
5
votes
1answer
2k views

Long exact sequence for a triple follows from long exact sequence for a pair?

In homology theory, the long exact sequence for a pair $(X,Y)$ is just $H(Y)\to H(X)\xrightarrow{\partial(X,Y)}H(X,Y)\to H(Y)[-1]$. The long exact sequence for a triple $(X,Y,Z)$ is $H(Y,Z)\to H(X,Z)\...
2
votes
2answers
106 views

Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$

Can we have some (new) examples of group extensions $G/N=Q$ with continuous (e.g. Lie groups) $G$ and $Q$, but a finite discrete $N$? Note that $1 \to N \to G \to Q \to 1$. What I know already ...
5
votes
0answers
67 views

Cohomology groups $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$

I'm trying to compute $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$. Here the semi-direct product is given by the outer automorphism of $PSU(4)$. By Serre spectral sequence, we have $...
2
votes
0answers
62 views

Cover and extension of a Lie group

We know that $SU(2)$ is a double cover of $SO(3)$, such that $$SU(2)/Z_2=SO(3),$$ through a finite extension $N=Z_2$. For other examples of simply-connected Lie groups such as $SU(2)$, $SU(N)$ or $...
8
votes
1answer
442 views

Contravariant Grothendieck Spectral Sequence

I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ (...
4
votes
1answer
182 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
3
votes
1answer
99 views

Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
1
vote
0answers
129 views

Bockstein homomorphism for $\mathbb{Z}_n$ and “Steenrod” $n$th power

The Bockstein homomorphism can be generalized for $\mathbb{Z}_n$ values, $$\beta_n: H^m(M^d,\mathbb Z_n) \to H^{m+1}(M^d,\mathbb Z_n),$$ and $$\beta_n x =\frac1n d x \text{ mod } n,$$ $$x \in H^m(M^...
1
vote
1answer
326 views

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 $\mathbb{Z}_2$ ...
1
vote
1answer
91 views

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 $$ 0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0 \,. $$ ...
5
votes
2answers
144 views

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 ...
5
votes
1answer
370 views

Are long exact sequences in homology a special case of spectral sequences?

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\...
2
votes
1answer
54 views

Contradiction in spectral sequence for $K(\mathbb{Z},3)$

$\newcommand{\Z}{\mathbb{Z}}$ Take the fibration $K(\Z,2) \hookrightarrow * \to K(\Z,3)$. Then $d_3^{0,2}$ is an isomorphism since this is the only way to get rid of $H^2(K(\Z,2))$ and to kill $H^3(K(...
0
votes
1answer
157 views

Partial Converse to “Pushout of a cofibration is a cofibration”

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I want a converse of this fact specialized to the case where I am pushing out a map BY a fibration: I.e., if I am given a diagram $ \begin{CD} E_1 @&...