# Tagged Questions

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### extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

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234 views

### stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

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### spectral sequence with non-trivial action on coefficients

Set-up:
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}_2\xrightarrow{p}\mathbb{Z}_2\rightarrow 1$$
Define ...

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### Hochschild-Serre spectral sequence and non-trivial action on coefficients

Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action ...

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### Differentials in the Lyndon-Hoschild-Serre Sequence for p=0

I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact ...

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330 views

### A question on some computation of group cohomologies

Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...

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530 views

### H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...

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### Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)

We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies ...

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479 views

### Transgression maps in group cohomology and group homology / duality of spectral sequences

I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem.
Let $G$ be a group, $H$ a normal subgroup of ...

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305 views

### Can the Bockstein spectral sequence be used to compute cohomology rings ?

If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...

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282 views

### Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?

The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...

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347 views

### Profiniteness Condition for Hochschild-Serre Spectral Sequence?

This question may seem elementary to experts but I am quite confused about it:
According to the entry of Lyndon–Hochschild–Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to ...

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395 views

### Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective

I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...

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### How to Compute Transgressions in a Serre Spectral Sequence?

For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...

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### Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...

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305 views

### Third bordism group of BG, where G is an arbitrary compact Lie group.

Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...

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### How to compute the cohomology of the general linear group with integral entries

Q: So how does one compute the cohomology groups $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$?
First note that $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$ is isomorphic to $H_B^*(Y/GL_n(\mathbf{Z}),\mathbf{Z})$ (Betti ...