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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5
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1answer
50 views

Why is $H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)$?

As the question title suggests, what is the easiest way to see that there is an isomorphism$$H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)?$$
1
vote
0answers
15 views

Restriction-Co Restriction Homomorphism

Let $G$ be a finite group and let $A$ be any $G$ module. Then it is well known that $H^n(G,A)$ is a subgroup of $\oplus_p H^n(G_p, A)$, where $G_p$ denotes a sylow $p$ subgroup of $G$. This is ...
1
vote
0answers
33 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
2
votes
1answer
25 views

Computing group cohomology in magma?

I don't how to construct a group module in MAGMA. Can someone show me how to compute group cohomology using MAGMA? For example, I am interested in the action of finite p-groups on abelian p-groups, ...
5
votes
2answers
43 views

When does a representation of $H\subset G$ on $V$ extend to a representation of $G$ on $V$?

Let $G$ be a finite group, $H$ a subgroup, and $\varphi:H\rightarrow GL(V)$ a finite-dimensional representation of $H$ over a characteristic zero, algebraically closed field. Let $\chi$ be the ...
2
votes
1answer
33 views

Completion and algebraic closure commutable

The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of ...
2
votes
0answers
26 views

What is Relator Matrix

At the third section of a paper "Computing second cohomology of finite groups with trivial coefficients" by G. Ellis et al., the authors write Suppose that $<\underline{x}\mid ...
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0answers
24 views

Making sense of the term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence.

I am having troubles understanding the superscript "G/N" in the third term of the standard inflation-restriction exact sequence $$ 0\to H^1(G/N,A^N)\to H^1(G,A)\to H^1(N,A)^{G/N}\to H^2(G/N,A^N)\to ...
2
votes
1answer
52 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
2
votes
1answer
43 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!
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0answers
43 views

Simplicial homology and cohomology of Möbius band

I'm trying to calculate simplicial homology and cohomology groups, with coefficients in $\mathbb{Z}$, of Möbius band $M$. I use triangulation given on the following picture: We have in this case ...
3
votes
1answer
51 views

group cohomology of 3-manifolds

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there ...
1
vote
1answer
31 views

Relations between $R^fG$ and either $\mathbb{C}^fG$ or $\mathbb{Z}^fG$.

Denote by $RG$ the group ring of the group $G$ over the commutative ring $R$. A result by Passman saying that if $R$ is a commutative ring then $$RG=R\otimes_{\mathbb{Z}}\mathbb{Z}G.$$ As a result, ...
2
votes
1answer
34 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
0
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0answers
31 views

Question about cohomology of free abelian group

Let $A$ and $B$ be finite abelian groups and suppose that $B$ acts on $A$. Now, suppose we have two surjective homomorphisms $f,g:\mathbb{Z}^n\twoheadrightarrow B$ for some $n\in \mathbb{N}$. This ...
3
votes
0answers
39 views

Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
3
votes
0answers
52 views

Schur multiplier of “large” groups.

Let $G$ be a finite group and let $M(G)=H^2(G,\mathbb{C}^*)$ be its Schur multiplier. For "small" groups I can compute the Schur multiplier by hand in terms of corresponding roots of unity. However, ...
2
votes
1answer
48 views

Can we see directly from the cocycle condition that 2-cocycles are symmetric?

Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
1
vote
1answer
30 views

Why is $\operatorname{Hom}_{\mathrm{Groups}}(G,A)$ isomorphic to $\operatorname{Hom}_{\mathrm{Ab}}(G/[G,G],A)$?

This question is inspired by an exercise from the Weibel's book on Homological Algebra (beginning of chapter 6 on Group Cohomology). Let $G$ be a group and $A$ be a $G$-module. My question simply is: ...
0
votes
1answer
12 views

Does $C < G$ imply $H_n(C,A) < H_n(G,A)$?

Suppose to have two groups $C$ and $G$ (not necessarily abelian) such that $C < G$ (subgroup, not necessarily proper). Let's fix an abelian group $A$ such that it is a trivial $G$-module (and ...
2
votes
1answer
101 views

The augmentation ideal of $\mathbb{Z}G$

Let $G$ be a cyclic group of order $p$ and let $IG$ denote the augmentation ideal of the group ring $\mathbb{Z}G$. I need to find $H^1(G,IG)$. Since $$0 \rightarrow IG \rightarrow \mathbb{Z}G ...
0
votes
1answer
18 views

Centralizer acting on the homology of a subgroup

Let $H\subset G$ be a subgroup. Let $E_*G$ be a free (right) $\mathbb ZG$-resolution of the trivial representation $\mathbb Z$. Because $E_*G$ is then also a free $\mathbb ZH$-resolution of the ...
4
votes
2answers
184 views

What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring ...
1
vote
2answers
81 views

Show that the categories $G$-mod and $\mathbb{Z}G$-mod are equivalent.

I have another basic question inspired from reading the sixth chapter of Weibel's "An Introduction to Homological Algebra". First version of the question: a bit ambiguous At the first paragraph, ...
0
votes
0answers
27 views

A clarification about the meaning of “Let $\mathbb{Z}$ be *the* trivial $G$-module”.

I have a question regarding a definition/lemma in the book from Charles A. Weibel, "An introduction to Homological Algebra". At page 161, there is a claim starting as follows: Let $A$ be any ...
2
votes
1answer
26 views

Non-amenability of $F_2$ via $l^2$-homology

Since the free group $F_2$ of rank $2$ is non-amenable, we have that $l^2$-homology of $F_2$ vanishes in degree $0$, i.e. $H_0(F_2,l^2(F_2))=0$. This means that for any $f\in l^2(F_2)$, the equation ...
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vote
0answers
39 views

Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
8
votes
1answer
76 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
18 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
131 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
23 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
32 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
77 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
58 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
93 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
34 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
55 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
vote
0answers
43 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
98 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
47 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
97 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 ...
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0answers
53 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
99 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 ...
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votes
0answers
26 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
14 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
56 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
61 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
101 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
81 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 ...