Talk:Group cohomology

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A passing comment from someone with no account: as I write (18th Nov 09) it says, in the non-abelian cohomology section:

"The first cohomology of G with coefficents in A is defined as above using 1-cocycles and 1-coboundaries."

That's not true. Knowing the set of 1-cocycles and the set of 1-coboundaries isn't enough. One needs an equivalence relation on 1-cocycles, and the H^1 is the equiv classes. In the non-abelian case the equiv classes can be of different sizes so you need to know the full equiv reln.

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—Preceding unsigned comment added by 155.198.192.80 (talk) 13:24, 18 November 2009 (UTC)

In the column "Formal construction", is there a "g_n+1" missing in the definition of d^n?

no.

The only reference link, Turkelli, Szilágyi, Lukács: Cohomology of Groups, is a broken link.

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The formula for d^n does look a little strange.

I think the g0 in front of the first term and the g_i in the second term are probably mistakes. 69.234.20.113 10:04, 16 March 2007 (UTC)

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Re the previous comment: the definition of the coboundary is correct for the inhomogeneous cochains seen in applications such as projective representations.

The article is, however, very confusing in its present form. The definition in terms of the Ext functor seems to produce homogeneous co chains, i.e. functions $G^{n}\to M$ obeying $\phi _{n}(gg_{1},gg_{2},\ldots ,gg_{n})=g\phi _{n}(g_{1},g_{2},\ldots ,g_{n})\,\!$ and with coboundary operator $\delta \phi _{2}(g_{1},g_{2},g_{3})=\phi _{2}(g_{2},g_{3})-\phi _{2}(g_{1},g_{3})+\phi _{2}(g_{2},g_{3})\,\!$ and so on. The description with the explicit formula for $\delta \,\!$ uses the inhomogeneous cochains defined as, for example, $\alpha _{2}(g_{1},g_{2})=\phi _{3}(1,g_{1},g_{1}g_{2})\,\!$ with its more complicated-appearing coboundary formula (which is correct as written though).

I'd love to rewrite it, but I am not an expert on Ext and Tor.

Mike Stone (talk) 13:36, 23 February 2012 (UTC)

I have now added a bit linking homogeneous and inhomogeneous chains

Mike Stone (talk) 15:27, 19 March 2013 (UTC)

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Does anyone know why the last equation of the "cochain complexes" section is so small? As this is where we're defining H^n it should be bigger, if anything. I looked at the code and it seems normal to me. Owen Jones

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Suggestions (I may add these myself if I feel energetic):

• The graded product (cup product, in the topological context)
• Kunneth formula
• Examples (e.g. the cohomology ring of an elementary abelian p-group, with coefficients in GF(p))

Jaswenso 02:31, 3 September 2007 (UTC)

Finite groups and further

I just added a sentence saying that the article deals only with finite groups for now (it may be in an awkward place...) It would be good to say something about at least profinite group cohomology, but to do it justice one would have to add quite a bit. RobHar 19:21, 8 September 2007 (UTC)

Sections needing references

I just added two "sections needing references" templates. The section on the relation to topological cohomology theories just requires a book. The section on the history makes several different claims and probably requires several references. For some of the statements regarding algebraic number theory, the current references (Serre and Milne) are probably enough. RobHar 19:37, 8 September 2007 (UTC)

character for R

The character for R in RP^\infty does not display properly. I would suggest a switch to tex. Tkuvho (talk) 04:30, 13 June 2010 (UTC)

Intro paragraph

Could someone (preferably someone more knowledgeable than me!) please rewrite the opening paragraph? As it stands it's not a good intro to the article. For example, "generalization to non-abelian coefficients" makes no sense in a context where no coefficients, abelian or otherwise, have been mentioned. TIA :-) Educres (talk) 15:18, 22 November 2011 (UTC)

I am even less knowledgeable than you, but I support your request. I believe that a reader who does not already know what group cohomology is, is unlikely to be able to figure it out from the opening paragraph. Maproom (talk) 23:19, 24 May 2012 (UTC)

I'm a professional mathematician and I cold not get anything out of this article. — Preceding unsigned comment added by 134.157.88.219 (talk) 16:33, 24 February 2014 (UTC)

I rewrote the first part of the lead to be more accessible, see if it helps. --Mark viking (talk) 22:30, 24 February 2014 (UTC)

Group extensions

No mention here of the use of low-dimensional cohomology groups to classify group extensions? This is a glaring omission, particularly in view of the historical development of homological algebra. For 5 points, why is it called the Ext functor? CFGrauss (talk) 23:27, 22 July 2015 (UTC)

The relation between central extensions and the second cohomology group are mentioned briefly. Most Wikipedia articles, including this one, are incomplete; please feel free to add more material on this subtopic. --Mark viking (talk) 05:09, 23 July 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Group cohomology/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs references and proper lead. Geometry guy 22:29, 31 May 2007 (UTC) Needs a discussion of restriction, inflation, transfer, corestriction. Needs a section on applications. RobHar (talk) 22:34, 16 December 2008 (UTC)

Last edited at 22:34, 16 December 2008 (UTC). Substituted at 02:09, 5 May 2016 (UTC)

The definitions of coboundaries in the Cochain Complexes section are incomplete

In the Cochain Complexes section, the definition of $d^{1}$ is missing. What is it? It would be needed for applying the definitions

$Z^{n}(G,M)=\ker(d^{n+1})$

and

$H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M).$

for $n=0$.

The definition

$\left(d^{n+1}\varphi \right)(g_{1},\ldots ,g_{n+1})=g_{1}\varphi (g_{2},\dots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}\varphi \left(g_{1},\ldots ,g_{i-1},g_{i}g_{i+1},\ldots ,g_{n+1}\right)+(-1)^{n+1}\varphi (g_{1},\ldots ,g_{n})$

has no sense in the case of $n=0$.

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My idea (based on [1]) is the following.

In order to extend the definition to be valid for $n=0$, we should regard the functions ${\tilde {\varphi }}:\{g_{0}\}\times G^{n}\to M:(g_{0},g_{1},...g_{n})\mapsto \varphi (g_{1},...g_{n})$ for $n>0$, and ${\tilde {\varphi }}=\varphi$ for $n=0$, where $g_{0}$ is the identity element of $G$, and $G^{0}:=\{g_{0}\}$. Then the general formula would be

$\left(d^{n+1}\varphi \right)(g_{1},\ldots ,g_{n+1})=g_{1}{\tilde {\varphi }}(g_{0},g_{2},\dots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}{\tilde {\varphi }}\left(g_{0},g_{1},\ldots ,g_{i-1},g_{i}g_{i+1},\ldots ,g_{n+1}\right)+(-1)^{n+1}{\tilde {\varphi }}(g_{0},g_{1},\ldots ,g_{n})$ , which yields $d^{1}(g_{1})=g_{1}\varphi (g_{0})-\varphi (g_{0})$.

But this looks a bit ugly. Could somebody formulate this a bit more nicely? — Preceding unsigned comment added by 89.135.18.105 (talk) 05:27, 29 January 2019 (UTC)

Article improvements

• Add computations of semi-direct products from Totaro - https://core.ac.uk/download/pdf/82201814.pdf
• Add computations of free products, weibel pg 170, give group cohomology of SL_2(Z)
• dihedral groups weibel 196-198
• group for riemann surfaces pg 205
• universal central extensions and long exact sequences

Wundzer (talk) 06:33, 23 October 2020 (UTC)

Notational mismatch?

The section Basic examples contains this sentence:

$H^{k}(C_{m},\mathbb {Z} )={\begin{cases}\mathbb {Z} /m\mathbb {Z} &k{\text{ even}},n\geq 2\\0&k{\text{ odd}},n\geq 1\end{cases}}$

But the "n"s on the right side of the equation do not occur on the left, and the "m" on the left side does not occur on the right.

I hope someone knowledgeable about this subject can fix this.50.234.60.130 (talk) 07:48, 6 December 2020 (UTC)

Whoops, that's a typo, thanks for catching that :) Wundzer (talk) 18:21, 7 December 2020 (UTC)