a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).

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6
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1answer
44 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 ...
0
votes
0answers
13 views

First cohomology of direct product (in the coefficients)

Let $k$ be a field and let $G = A \times B$ be the product of two algebraic groups over $k$ ($G$ is not necessarily finite nor abelian). Is there a nice way to express $H^1(Gal(k^s/k), G(k^s))$ in ...
8
votes
0answers
121 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
0
votes
0answers
19 views

Induced homology morphism of invertible linear transformation

I'm doing some excercises from Hatcher. I'm dealing with excercise 7 in section 2.2 (page 164 in PDF file): For an invertible linear transformation $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ show that ...
1
vote
1answer
28 views

Existence of non-abelian split metacyclic extensions.

Is there any necessary conditions that must be held in order to guarantee the existence of a non-abelian split metacyclic extension? i.e. for which $m,n\in\mathbb{Z}$ there exist a non abelian split ...
1
vote
1answer
43 views

is the homomorphism induced by the inclusion map is the inclusion map?

let $X$ be a topological space, $A\subset X$ subspace. Consider $i:\:A\to X$ the inclusion, and $i_* :\:H_n(A)\to H_n(X)$ the induced homomorphism. is $i_*$ the natural homomorphism $[a]\mapsto ...
0
votes
1answer
57 views

Why is the inclusion an isomorphism?

Consider $X$ a path-connected space, $A\subset X$ a non-empty subset. My textbook makes the following claim without any explanation, and I wondered if you could help: it says that the inclusion $H_0 ...
2
votes
1answer
72 views

Prove or disprove any continous map $f$ from $T^2$ to $RP^2$ is null-homotopic.

I tried to solve the following question: Prove or disprove any continous map f from $T^2$ to $RP^2$ is null-homotopic. We know the universal cover of $RP^2$ is $S^2.$ I want to construct a map $g$ ...
1
vote
1answer
25 views

Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
2
votes
1answer
54 views

Calculating $H_1(\mathbb{R})$

Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't ...
1
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0answers
29 views

Group cohomology of free abelian groups

I am looking for a good reference for the structure of the cohomology ring $H^*(Z^n,Z)$. In particular, I would like to know how large is the subgroup of $H^2(Z^n,Z)$ generated by cup-products from ...
4
votes
1answer
95 views

Values of the Herbrand quotient

For a finite cyclic group $G$, there is the Herbrand quotient in the theory of group cohomology. I calculated some of those quotients and I always came up with an Integer as solution. I failed at ...
2
votes
1answer
45 views

Order of elements in group cohomology

OK, I'm just learning about cohomology of groups, and I want to make sure I'm not missing anything in this question: Let $n$ be an abelian group, let $G$ be a group, let $\tau$ be an action of ...
5
votes
0answers
68 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
1
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0answers
38 views

Shapiro's Lemma-Finding the inverse of an isomorphism.

Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is $\phi(f+B^n(G,Hom_{ZH}(ZG, A) ...
2
votes
0answers
93 views

How to compute/ find cancellation for the second group cohomology $H^2(G,A)$?

My problem is the following, suppose you have a discrete group $G$ (finite type) and a $G$-module $M$, $M$ is a $\mathbb{Q}$-vectorial space. I would like to know if there are "ways" to compute ...
0
votes
0answers
21 views

Show $X$ is a H-space?

Let $Y$ be a H-space and suppose $X$ is a pointed retract of $Y$ with continuous pointed maps $s,r$. My thinking so far; If Y is a H-space then there is a map $m:Y$x$Y \rightarrow Y$ such that $m$ ...
0
votes
0answers
11 views

Cochain boundary operator identity question (Morandi 2.10.3)

This is problem 2.10.3 From Morandi's Field and Galois Theory: Let $M$ be a $G$-module, and let $f \in Z^2 (G,M)$. Then: $$f(1,1) = f(1,\sigma) = f(\sigma, 1) \space \forall g \in G$$ I can get ...
2
votes
1answer
54 views

What kind of cohomology is meant?

What kind of cohomology is meant in Deligne's work about mixed hodge structure on cohomology groups of an complex algebraic variety? I think it refers to the singular cohomology with coefficients in ...
0
votes
1answer
55 views

hom functor exactness and group cohomology

On one hand we have this lemma: let $A_1\rightarrow A_2 \rightarrow A_3\rightarrow 0$ be exact sequence of G-modules. Then $0\rightarrow \text{Hom} _G(A_3,M)\rightarrow \text{Hom} _G(A_2,M) ...
4
votes
2answers
96 views

can we derive integral cohomology from rational cohomology and mod p cohomology?

Let $X$ be a topological space. If we know that for $\mathbb{F}=\mathbb{Q}$ and $\mathbb{Z}/p$, for any prime $p$, $$ H^*(X;\mathbb{F})=0$$ for any $*\geq n+1$, can we conclude that $$ ...
0
votes
0answers
15 views

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$?

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$? $H^{1}(K)$ is the first de Rham cohomology group of $K$.
3
votes
1answer
69 views

On a property of split short exact sequences

Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short ...
0
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2answers
53 views

Tate-Shafarevich groups and Hasse principle (reference)

I'm looking for a proof of the fact that the Hasse local-global principle holds for an elliptic curve $E$ defined over $Q$ if and only if the Tate-Shafarevich group of $E$ vanishes. I just need to ...
2
votes
1answer
60 views

Is the group cohomology for a profinite group always torsion?

In his notes on group cohomology here, Bjorn Poonen claims that $H^i(G, A)$ is torsion when $G$ is profinite and $i>0$. why is the following not a counterexample? Take $G= \hat{\Bbb Z}$, and ...
1
vote
0answers
30 views

Independent components of a group cocycle

Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all ...
3
votes
2answers
47 views

Minimal presentations and (co)homology groups

I wonder whether there exists a link between the number of generators and relations of a presentation for a given group $G$ and the ranks of its (co)homology groups $H_1(G,\mathbb{Z})$ and $H_2(G, ...
2
votes
0answers
34 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
1
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0answers
29 views

Functoriality of the cup product

This is a basic question about the meaning of functoriality of cup products. On page 105 of "Cohomology of Groups" appearing in Cassels and Frohlich's Algebraic Number Theory, Atiyah and Wall define ...
0
votes
0answers
16 views

Definitions of the group of cycles/group of boundaries

first I want to clarify that the class I am referring to is not really about homological algebra, rather about Galoistheory. Still we defined the group of cycles/ the group of boundaries (first ...
1
vote
1answer
75 views

Homology group $H_1(G;\mathbb{R})$ is a vector space?

I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I ...
0
votes
0answers
29 views

Prerequisites to study cohomology?

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make ...
1
vote
1answer
69 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
1
vote
1answer
64 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
1
vote
1answer
37 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
1
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0answers
36 views

Hochschild-Serre spectral sequence for not normal subalgebra

I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here ...
1
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0answers
35 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
2
votes
0answers
29 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
0
votes
0answers
19 views

let $G$ be a group with $cd(G)=m$ ,let $U$ be a subgroup of $G$ of finite index in $G$ ,show that $cd(U)=m$ .

let $G$ be a group with $cd(G)=m$ and $U$ be a subgroup of $G$ of finite index in $G$. Show that $cd(U)=m$ . $cd(G)$:a group $G$ has cohomological dimension$\leq n $ ,denoted by $cd(G)\leq n $ if ...
0
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0answers
21 views

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$.

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$. any hint or idea or references to study will be great,thanks.
2
votes
0answers
28 views

if $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$.

If $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$. It will be great if you help me with this. Any hint or guidance will be great. Thanks.
3
votes
2answers
120 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
4
votes
1answer
74 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
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0answers
46 views

Cohomology calculation or computation?

I have a terminology question: Does one compute the cohomology of a group, or does one calculate it? Is it more common to speak of cohomology calculation or cohomology computation? Thanks for your ...
2
votes
1answer
64 views

Proof for a theorem on Cohomology by Tate

I am searching for a reference for the proof of the following theorem. Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = ...
3
votes
0answers
38 views

Group cohomology via resolutions

I have a basic question about the definition of group cohomology. Suppose $\Gamma$ is a discrete group, $R$ a commutative ring and $V$ is an $R\Gamma$ module. Then, the first definition of group ...
1
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0answers
50 views

Definition of module

In the book's, The theory of numbers, S. Iyanaga. Chater I, Cohomology of groups. What is the meaning of "module A"? Thank you all.
1
vote
2answers
93 views

Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
5
votes
1answer
79 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
2
votes
1answer
104 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...