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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21 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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20 views

When does the first cohomology group commute with inverse limit?

Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
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1answer
39 views

Absolute Galois group $\text{Gal}(\overline{K}/K)$ of any number field $K$ has a non-open subgroup of any prime index $p$?

Let $K$ be a number field, and let $p$ be a prime number. Does $G = \text{Gal}(\overline{K}/K)$ necessarily have a subgroup of index $p$ that is not open?
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1answer
32 views

Easy computational example of first cohomology group: Is this how we do it?

I'm an undergraduate student learning a little bit of group cohomology on my own. I'd like to compute a few examples of the low-dimensional cohomology groups in some special cases to get some ...
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9 views

Notation in group cohomology context

Let $M$ be a $G$-module and let $\psi\colon G\to M$ be a map. What is usually meant with the symbol $\psi_{\sigma}$ where $\sigma\in G$ in the group cohomology context? I am using the appendix from ...
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24 views

For any closed form $a$ with compact support, there exists a form $b$ w.c.s. in the unit ball such that $a-b$ is exact.

Let $\alpha$ be a closed (differential) $k$-form with compact support in $\mathbb{R}^{n}$. We want to prove that there exists a $l$-form $\beta$ with compact support in the unit ball of ...
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53 views

Let $G$ be a group. Why is $ \operatorname{Ext}_{\mathbb{Z}G}^1(\mathbb{Z},\mathbb{Z})\cong \operatorname{Hom}_{Grp}(G,\mathbb{Z})$?

Let $G$ be a Group and $\mathbb{Z}G$ is the ring of the formal sums $$\sum_{g\in G}n_gg$$ with multiplication $$(\sum_{g\in G}n_gg)(\sum_{h\in G}m_hh)=\sum_{g\in G}(\sum_{h\in ...
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29 views

How to prove $M\otimes_{\,G}A≅ {Hom}_{G}({Hom}_{\,\mathbb Z}(M,{\,\mathbb Z}),A)$?

Given that G is a finite group, M is a finitely generated right free G-module and A is a left G-module, there exists a natural G-isomorphism $\phi\ : M\otimes_{\,G}A\rightarrow ...
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1answer
46 views

Cartesian product of compact triangulated spaces

Let $X$ and $Y$ two compact triangulated spaces, I am trying to show that $X\times Y$ is also a compact (this is obvious) triangulated space and $$\chi(X\times Y)=\chi(X)\cdot\chi(Y)$$ Any tips on ...
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1answer
26 views

How to prove $\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}G,A)^{S}\cong\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}(G/S),A)$?

Given that S is a normal subgroup of a finite group G and A is a G-module, I have difficulty in proving $\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}G,A)^{S}\cong\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}(G/S),A)$. Would ...
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1answer
9 views

The boundary formula and cohomology of finite groups

I've a very basic notational question on group cohomology. Let $G$ be a finite group and $M$ a $G$-module. For $i\geq 0$, let $P_i=\mathbb Z[G^{i+1}]$ be the free $\mathbb Z$-module on $G^{i+1}$, made ...
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1answer
31 views

An odd expression appearing in proof that kernel and image of certain map on group ring $\mathbb Z[G]$ are equal

Let $G = \langle g \rangle$ be a cyclic group of order $n$. Consider the free ring $\mathbb Z[G]$ of all formal sums of elements from $G$ with coefficients from $\mathbb Z$, with multiplication given ...
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17 views

Locally cyclic group and homology

I would like to ask about the accuracy of the following statement. If $G$ is locally cyclic. Then the $n$-th homology group of $G$ $($H_n(G) $)$ (specially for $n=2$, $n=3$) is also locally cyclic. ...
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34 views

Dimension of $\text{H}^n(G, KG)$

I was reading the paper "The second cohomology group of $G$ with $Z_2G$ coefficients" by Thomas Farrell. I came across the following question which was asked by Serre in 1974: Let $G$ be a ...
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41 views

Augmentation ideal and the abelianization of $G$

On a qual problem recently, I came across the following fact: If $G$ is a finite group, and $\mathfrak{a}$ is the augmentation ideal of the integral group ring $\mathbb{Z}G$, then ...
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47 views

Historical notes on the Jordan-Hölder program

I'm looking for any material (books, articles..) documenting the historical process of the formulation, partial work and/or the actual stage of the Jordan-Hölder program. I'm not sure if there is any ...
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1answer
52 views

Calculation of Group Cohomology of $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Z}$

I am trying to learn some group cohomology and I'm starting to get my head around the theory, but I find it hard to find some explicit examples of the calculation of group cohomology of some small ...
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196 views

Which groups act freely on $S^n$?

When $n$ is even, it is easy to classify groups which act freely on $S^n$ using degree theory: if $G$ acts on $S^n$, then associating to each element $g \in G$ the degree of the map obtained from ...
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30 views

Group cohomology as a right derived functor

In this Wiki page, it says that group cohomology can be defined as right derived functor of $F$, where $F(M)=M^G$. There are two different equivalent definition in the page, by explicit cochains and ...
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91 views

I want to show the group cohomology $H^{n>1}\left(F,\,M\right)$ vanishes whenever $F$ is free.

I want to show the group cohomology $H^{n>1}\left(F,\,M\right)$ vanishes whenever $F$ is free. I tried to show $\text{pdim}_{\mathbb{Z}\left[F\right]}\left(\mathbb{Z}\right)\le 1$, but we know ...
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19 views

Open subgroups $V\subset U$ of profinite group such that the index $[U:V]$ is divisible by $p$.

Let $G$ be a profinite group of $p$-cohomological dimension, $cd_p(G)\neq 0,\infty$. Consider an open subgroup $U\leq G$. Does there exist an open subgroup $V\leq U$ such that its index, $[U:V]$ is ...
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67 views

Bourbaki's definition of semidirect product

This recess I'm off to learn about group extensions and the cohomological methods to characterize those extensions, but I'm a bit stuck on wraping my head around all the new concepts (I just finished ...
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1answer
32 views

Determining if a (Lie algebra) central extension is trivial.

Given a central extension for a given Lie algebra, is there any simple way to check that it is/isn't isomorphic to the trivial extension ("simple" meaning, not as tedious [and daunting, for an algebra ...
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1answer
25 views

Reference for a result

A friend of mine told me that the cohomology of $\pi_1(M)$ was isomorphic to the cohomology of the manifold $M$. Is that true (maybe there are some hypothesis) ? Does someone know a reference for this ...
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1answer
19 views

$G$ acts $\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}G,A)$

Let $A$ be a $G$-module where the action $G$ on $A$ is trivial. Then also $G$ acts on $\rm{Hom}_{\,\mathbb Z}(\mathbb{Z}G,A)$ such that $g\cdot f(x):=f(xg)$. On the other hand, we know that ...
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1answer
31 views

Reference (or proof) of: classifying map $u\colon M \to K(\pi,1)$ induces $H^1(\pi_1(M);\mathbb{Z}/2)\cong H^1(M; \mathbb{Z}/2)$

I'm try to understanding a survey chapter in Angeloni - Metzler -Sieradski "Two dimensional Homotopy and combinatorial group theory" namely the one about Stable classification of $4$-manifold (see ...
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35 views

Unramified cocycles and the Selmer group of an ellptic curve

In Silverman's book on elliptic curves, he gives a procedure to compute the Selmer group of elliptic curve $E$ relative to an isogeny $\phi:E\to E'$. I am confused about one step in the discussion. ...
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50 views

Third cohomology group of abelian groups

What is the third cohomology group $H^3(C_n,\mathbb{Q}/\mathbb{Z})$ of the cyclic group $C_n$ of order $n$? How can I calculate the third cohomology group of finite abelian group $A$ with ...
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25 views

Poincaré Duality as a $G$-morphism

In Brown's Cohomology of Groups book, at page 211 there is the following statement (Prop. 8.2): Let $Y$ be a compact $d$-dimensional $K(G,1)$-manifold (possibly with boundary). Let $X$ be its ...
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1answer
40 views

Why are projective representations of a group classified by the second cohomology group?

I'm reading about the classification of bosonic SPT's, and I came across this statement: projective representations, where $v(g_1)v(g_2)=\alpha(g_1,g_2)v(g_1g_2)$, $v(g_1)$ being the transformation ...
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1answer
42 views

Difference between $\mathbb{Z} G$-module and $G$-module

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is ...
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2answers
49 views

Compute the first group cohomology with trivial $G$-modules

This is the exercise 6.1.5 in Weibel's book. Let $G$ be any group and $A$ be a trivial $G$-module. Show that $H^1(G;A)\cong\hom(G,A)\cong\hom(G/[G,G],A)$ In the book, $H^*(G;A)$ is defined to be ...
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1answer
76 views

Cohomology result (reference request)

I want to have reference for this result. Let $G$ be a group of order $p^k$ where $k\geq0$ and $A$ be a $G$-module. If for all positive $r$ and for all subgroup $H$ of $G$, $$H^r(H,A)=0$$ then for ...
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1answer
31 views

Cohomology of closed subgroup of profinite group

I have been reading Cassels-Frohlich, and I have a question about a fact that is cited in a proof about the cohomology of pro-finite groups (the exact palace is the proof of Prop 4 of section 2.8 of ...
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30 views

What is the Cohomology Tree Probability Distribution?

As the title says, I'm interested in finding out what is meant by Cohomology Tree Probability Distribution. I came across this term on group props, and it seems to be a distribution for randomly ...
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36 views

Reference on Weibel's Homological Algebra: “$G/H$ acts by conjugation in LHS-spectral sequence”

I'm studying the Lyndon-Hochschild-Serre spectral sequence for $H\triangleleft G$: $$ H_p(G/H;H_q(H;A))\Rightarrow H_{p+q}(G;A) $$ where $A$ is a $G$-module. I was told (w/o giving a proof) that ...
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1answer
16 views

Cohomological dimension of $G$ equals $0$ iff $G=\{0\}$

I want to prove this statement with elementary considerations about group cohomology (started studying it today) If $G$ is a group such that its cohomological dimension is $0$, then $G$ is the ...
3
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1answer
43 views

Notation of a $G$-module in a group cohomology paper

In the first page of the paper titled "The mod $p$-cohomology ring of $\operatorname{GL}_3(\mathbb{F}_p)$" (found here), there are two notational questions that pop up for me. I'll try to reproduce ...
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31 views

write down a concrete 2-coboundary

Let $G=SL_3(\mathbb{Z})$, the group of 3 by 3 matrices with determinant 1. Then by a deep theorem of Borel and Serre proved in this paper "Corners and arithmetic groups", we know that the 2nd ...
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52 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
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140 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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20 views

Prove that $ \text{H}^1(G/P,\text{Z}(P)) = 0 $ for a normal Sylow p-subgroup P

Let $ p $ be a prime and $ G $ a (multiplicatively written) finite group with a normal Sylow $ p $-subgroup. Let $ Q = G/P $. Then $ Q $ acts on $ \text{Z}(P) $ via conjugation: $ gP.z = gzg^{-1} $. ...
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76 views

Group cohomology: $\mathbb{Z}G$-maps from $G^{n+1}$ are the same as set maps from $G^n$

I'm learning about group cohomology from Knapp's book Advanced Algebra. Given a group $G$, and an abelian group $M$, he defines $F_n$ to be the $(n+1)$-fold product of $G$, endowed with a $G$-action ...
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23 views

Does the Bockstein commute with maps induced by group homomorphisms?

Let $G$ be a finite group, and let $\sigma:G\to G$ be a homomorphism of groups. There is an induced map in cohomology $\sigma^*:H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$. We also have the ...
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1answer
37 views

Conjugation action of $G$ on $N$ induces an action of $G/N$ on $H_{\ast}N$.

I need some help in proving this result from Weibel or Brown (page $48$ corollary $6.3$). If $G$ is a group and $N$ a normal subgroup of it, then the conjugation action of $G$ on $N$ induces an ...
3
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1answer
31 views

What short exact sequence induces the Bockstein for $H^*(G,k)$?

Let $G$ be a finite group, and let $k$ be an algebraically closed field of characteristic $p$. An element $z\in H^n(G,k)$ can be represented by an $n$-fold extension of $k$ by $k$ in the category of ...
3
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1answer
32 views

A nonsplit extension of a nonabelian finite simple group by a cyclic group of odd prime order

Let $p$ be an odd prime. Does a nonabelian finite simple group $S$ exist such that $H^2(S, \mathbb{Z}/p\mathbb{Z})$ is not trivial?
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1answer
26 views

Tuples of cohomology classes coherently related via restrictions and conjugations

I'm trying to parse a statement I've found in The Cohomology of Groups by Leonard Evens. For notation, let $G$ be a finite group, let $k$ be an algebraically closed field of characteristic $p>0$, ...
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0answers
35 views

Group cohomology of $\mathbb{R}^\times$

Consider topological group $\mathbb{R}^\times$ with a standard topology and $\mathbb{R}^\times$-module $S'(\mathbb{R})$ consisting of all Schwartz distributions on $\mathbb{R}$ where ...
4
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1answer
66 views

how to calculate $H_2(S^{2}\times S^{1}\# S^{2}\times S^{1}\# S^{2}\times S^{1})$ ?

When advancing in some calculations, I found the problem of computing: $H_2(S^{2}\times S^{1}\# S^{2}\times S^{1}\# S^{2}\times S^{1})$. I found this Mayer-Vietoris sequence of which is: $$0\to ...