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Results tagged with group-cohomology
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user 318
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
4
votes
The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for ...
Some further data points:
(2') For a finite field $\mathbb{F}_{p^r}$ with $p^r \neq 2$ one has $\mathbf{N}(\mathbb{F}_{p^r}, j) \leq \max(\lceil\tfrac{j}{2}\rceil, j-r(p-1)+3)$.
This is by combining T …
- 18.2k
6
votes
2
answers
2k
views
Transfer homomorphisms with coefficients
In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^* (H;M) \to H^*(G;M)$.
In homotopy theory, there is a transfer ma …
- 18.2k
5
votes
Accepted
Cohomology of SL(2,R) with coefficients given by linear action
It is zero. This is an application of the "centre kills" trick, which I will state in homology.
Trick. Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1 …
- 18.2k
1
vote
Accepted
Transgression in terms of k-invariant for chain complexes
I think the difficulty is that you are assuming that $X$ only has homology in two degrees, but are then looking at the cohomology spectral sequence. (To get sensible answers I seem to have to take coh …
- 18.2k
13
votes
Accepted
Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?
Let $G_n := W(\tilde{A}_{n-1})$. If I understand your description correctly, there is an extension
$$1 \to G_n \to S_{n} \wr \mathbb{Z} \overset{sum}\to \mathbb{Z} \to 1$$
and so a $\mathbb{Z}$-Galois …
- 18.2k
6
votes
Accepted
Naturality of the transfer in group cohomology
I don't believe this is true. Let $(G, H) = (\Sigma_3, C_3)$ and $f : C_3 \to \Sigma_3$. Then your square says that
$$H^1(C_3;\mathbb{Z}/3) = \mathbb{Z}/3 \longrightarrow H^1(\Sigma_3;\mathbb{Z}/3) = …
- 18.2k
1
vote
Equivalence of finiteness of $spliG$ and periodicity isomorphisms being induced by cup product
Let $P_\bullet \to \mathbb{Z}$ be a projective resolution as a $\mathbb{Z}[G]$-module, $\Delta : P_\bullet \to P_\bullet \otimes P_\bullet$ an approximation of the diagonal, and $\phi : P_\bullet \to …
- 18.2k
19
votes
Accepted
Is the cohomology ring of a finite group computable?
As I understand it this follows from Benson's Regularity Conjecture, proved by Symonds fairly recently. It says that $b_p = 2(|G|-1)$ will do.
- 18.2k
6
votes
1-st cohomology of multiplicative group in a vector space
By coincidence I needed to know something about this recently, and one thing I know is that that Ext group vanishes for $0 < \vert m - n \vert < p - 1$.
There is a proof in Lemma 6.1 of my paper "Co …
- 18.2k
6
votes
Homology of a limit of semidirect products
No, because it is not even true for constant families: let $A$ be an acyclic group, so $H_i(A)=0$ for $i>0$, and $B$ be a group which $A$ acts on interestingly, e.g. $B= F(A)$ is the free group on the …
- 18.2k
9
votes
Accepted
Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2...
The group $U(1) \rtimes \mathbb{Z}/2$ you describe is nothing but the group $O(2)$ (as $U(1) = SO(2)$).
As such I think one can see the spectral sequence for the extension does collapse, and one obta …
- 18.2k
7
votes
Accepted
Is there a kind of Poincare duality for Borel equivariant cohomology?
This kind of thing shows up quite naturally in parameterised stable homotopy theory. Let me translate an idea I know from there into the language in this question.
Cap product gives a map
$$C^{p}(M …
- 18.2k
10
votes
Accepted
How can I detect the homology image of a unipotent group in the general linear group?
Suppose first that $F$ is a finite field of characteristic $p$. Then $U_n(F)$ is a Sylow $p$-subgroup of $GL_n(F)$, and so using the transfer in group homology one sees that the image of $f_k$ (for $k …
- 18.2k