Questions tagged [group-cohomology]
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.
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Is there a simple explicit expression of the Pontryagin square in terms of the cup product on a spin 4-manifold?
$A$ a finite abelian group, and denote $\Gamma(A)$ its universal quadratic group. The Pontryagin square $\mathfrak{P}\in H^4(B^2A,\Gamma(A))\cong \text{Hom}(\Gamma(A),\Gamma(A))$ is the element ...
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Cohomology of $\operatorname{GL}_3(\mathbb{F}_2)$
I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in ...
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Group cohomology of ${\rm SL}_{n}(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$
Let $p>5$ be a prime. For any integer $n\geq 2$, let $M_{n}^{0}(\mathbb{F}_p)$ denote the $n\times n$ matrices with trace $0$ over a finite field $\mathbb{F}_p$ of order $p$. Then we have an action ...
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Ker of corestriction of Galois cohomology
Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, ...
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Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products
The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as
$$
\rho_1(a,b)=(b,a) \ .
...
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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Is there a strictly coassociative resolution of polynomial growth, for a finite group?
Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
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Are these two natural cohomology classes of a manifold constructed from a 1-cochain and a group extension equal?
Let $X$ be a manifold, $G$ and $A$ finite abelian groups and $\epsilon \in H^2(G,A)$ a group cohomology class (for the moment I am assuming there is no action of $G$ on $A$). Given $\alpha \in H^1(X,G)...
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Cohomology ring $H^*(\operatorname{SL}(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$
$\DeclareMathOperator\SL{SL}$In Soulé's paper "The cohomology of $\SL_3(\mathbb{Z})$" the cohomology ring $H^*(\SL(3,\mathbb{Z}),\mathbb{Z})_{(2)}$ is determined in Theorem 4.iv. I'm wanting ...
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A Tate resolution for $\Sigma_p$ - Reference request
Below I will describe a mod $p$ Tate resolution for the symmetric group $\Sigma_p$, i.e. a $\mathbb{Z}$-graded periodic acyclic chain complex $C^*$ of finitely generated modules over $\mathbb{F}_p[\...
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Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
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A map in group cohomology from $H^n(G,G^{\vee})$ to $H^{n+1}(G,U(1))$
Let $G$ be a finite abelian group and denote by $G^{\vee}=\mathrm{Hom}(G,U(1))$ its Pontryagin dual. For any positive integer $n$ one can define a homomorphism of abelian groups
$$
f:H^{n}(G,G^{\vee})\...
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Example of continuous cohomology vs cohomology
I am looking for an example of a locally compact group $G$ and a continuous $G$ module $M$, which also is locally compact, such that the continuous cochain cohomology differs from group cohomology (...
user473423
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Groups acting on categories produce 2-cocycles
$\DeclareMathOperator\Hom{Hom}\newcommand\id{\mathrm{id}}\DeclareMathOperator\Aut{Aut}$Let $\mathcal{C}$ be a category (such that each hom sets are $\mathbb{C}$ linear spaces) and $G$ be a group. We ...
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Determine if a 2-cocycle is zero in $H^2(G,\mathbb C^\times)$
Let $G$ be a finite group with trivial action on $\mathbb C^\times$. And given a 2-cocycle $\alpha$ in its Schur multiplier group $H^2(G,\mathbb C^\times)$, as an explicit map from $G\times G\to \...
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Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
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Cohomological finiteness (boundedness) property
Let $G$ be arbitrary group. Let us assume it is $\operatorname{FP}_\infty$. Suppose that the integral cohomology groups $H^i(G, \mathbb{Z})$ have bounded rank as finitely generated free abelian groups ...
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Cohomology of cocompact lattices in hyperbolic spaces
I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
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Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve
[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
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Explicit formulas for some homotopies - already known?
Let $G$ be a group and let $F\xrightarrow\varepsilon\mathbb Z\to 0$ be the corresponding standard resolution, with $F_n=G^{n+1}$ and $\partial (s_0,\ldots,s_n)=\sum_{i=0}^n(-1)^i(s_0,\ldots,\hat s_i,\...
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Computation of $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$
As a term of a Serre spectral sequence, I would like to compute the cohomology group with compact support $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$ of the moduli space of genus $1$ curves with $2$ ...
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Algebraic models of cohomology classes of (higher) Eilenberg-Maclane Spaces?
In Classification of weak 3-groups, Qiaochu gave an excellent answer, in which, he mentioned cohomology classes $H^{4}(B^{2}\pi_{2};\pi_{3})$ can be viewed as quadratic refinement of Whitehead bracket ...
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Two results on cohomology adapted to cochains
Given $G$ a group and $M$ a $G$-module, we denote by $(C(G,M),d)$ the cochain complex resulting from the standard resolution. An element in $H^n(G,M)$ can be written as the class $[a]$ of an element $...
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Additivity of group cocycles?
In Juven Wang, Zheng-Cheng Gu, and Xiao-Gang Wen - Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, the authors calculated many ...
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“Sheaf cohomology” of Galois groups
Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...
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Torsion in the first cohomology of a lattice in a semisimple Lie group
Let $\Gamma$ be a cocompact lattice in a complex semisimple Lie group $G$ of dimension $n$. Let $M$ be a $\mathbb{Z}\Gamma$-module, finitely generated as an abelian group (let $r$ be the minimal ...
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Fibrations and Euler characteristics with bad fundamental group
Consider a fibration $F\to E\to B$ where $H^i(F;\mathbb{Q})$ and $H^i(B;\mathbb{Q})$ are finite-dimensional, and they vanish for $i\gg 0$, and $B$ is connected. However, we do not assume that $B$ is ...
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If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?
$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some ...
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Identifying group extension from cohomology class of $D_8$
I have the following problem. It is well known that $H^\ast(D_8,\mathbb{Z}/2)\cong \mathbb{F}_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). ...
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What is this cochain complex about, whose $H^1 = \mathbb{R}$?
$\DeclareMathOperator\QEnd{QEnd}$Let $C^n$ be the set of functions $\mathbb{Z}^n \to \mathbb{Z}$, and $B^n$ the set of bounded such functions. For $a_1,...,a_{n+1} \in \mathbb{Z}$, the differential of ...
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Abelianisation of certain congruence subgroups in GL_2(Z)
$\DeclareMathOperator\SL{SL}\newcommand{\ab}{\mathrm{ab}}$Denote by $$\Gamma(m) = \left\{ \left( \begin{array}{cc} 1 +ma_{11} & m a_{12} \\ ma_{21} & 1 +ma_{22} \end{array} \right) \in \SL_2(\...
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Trivial homology groups for p-torsion groups
Let $G$ be a group where each element has a $p$-power order.
Let $M$ be a $G$-module without $p$-torsion.
Here $G$ is a discrete infinite subgroup of a complete group. Then, it cannot be assumed pro-...
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Cohomology of modular curve
(A follow-up on this). Consider the modular curve $X_0(N)$. I'm trying to make the jump from understanding the cohomology $H^1(X_0(N), \mathbb{Z})$ to understanding $H^1(X_0(N), \mathcal{O})_\mathfrak{...
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Context for Wiles defect criterion and patching
This is not a homework or a project question, just me trying to get acquainted to the subject: I am an MSc student who recently came across the Wiles defect numerical criterion (see, for example, ...
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Comparing cohomology of a total complex with the cohomology of semidirect product
$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
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What's the use of group cohomology for class field theory?
I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...
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What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...
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Question on a theorem about group extensions
I am reading the chapter Second cohomology groups of Continuation of the Notas de Matemàtica.
Part 1 in theorem 1.2 tells us that
Let $E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1$ be an ...
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What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with perfect fundamental group?
In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of ...
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Projective dimension of group ring
Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...
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Cohomology of a countable directed union of groups
It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...
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What properties do the categories $\mathbf{GrpMod}$ and $\mathbf{GrpMod}^*$ of compatible pairs have? Can we do homological algebra with them?
Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is:
an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ ...
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Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle
of $G=\GL_{n,\R}\,$, that is,
an invertible ...
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Explicit cochain for Shapiro's lemma with trivial coefficients
(cross-post from stack exchange after not receiving any answers)
I'm wondering the following: if we have a finite-index subgroup $H\subset G$, and a cocycle $[c]\in H^1(H,\mathbb{Z})$, is there any ...
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Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
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Do algebraic tori have no $H^1$?
If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...
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Cohomological variety in case that Sylow subgroup is elementary abelian
Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V_G$ of ...
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Finite domination and Poincaré duality spaces
Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...
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Special groups, special resolutions and group cohomology
$\newcommand{\Z}{\mathbf{Z}}$ Let $G$ be a non-abelian group. And let $\Z$ be the ring of integers.
Under which condition on the group $G$ can we find a free resolution $F_{\bullet}\rightarrow \Z$
of $...
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Extensions of a simple group by an elementary abelian $p$-group
Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other ...
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