# Essential theorems in group (co)homology

I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of

1. Hopf's formula - If $G$ has presentation $F/R$, then $H_2(G)=R \cap [F,F]/[F,R]$
2. If $G$ has torsion then $H_n(G)$ has no top dimension
3. $H_n = Tor_n$ so is the left derived functor of $\otimes$
4. $H^n = Ext ^n$ so is the right derived functor of $Hom$
5. If $G$ is discrete, then $H_n(G)=H_n(K(G,1))$
• What does "top dimension" mean in group cohomology? – darij grinberg May 25 '10 at 11:49
• Homology groups are all trivial for $i>n$ if $n$ is the top dimension. – Josh May 26 '10 at 0:41

The Hochschild-Serre spectral sequence

• Should have listed that one! I've been fighting with some differentials from that spectral sequence lately. – Josh Jan 21 '10 at 14:30

Cameron Gordon's theorem that $H_2(G)$ is generally non-computable from a group presentation of G. IMO this should be the standard appended caveat to Hopf's formula.

Gordon, C. Some embedding theorems and undecidability questions for groups. Combinatorial and geometric group theory (Edinburgh, 1993), 105--110, London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995.

• It would be way more economical to append standard caveats to theorems whose results are computable :) – Mariano Suárez-Álvarez Jan 21 '10 at 16:50
1. Interpretation of cohomology of small degree:

$H^1(G,A)$ = crossed homomorphisms $G\to A$ modulo principal ones.

$H^2(G,A)$ = equivalence classes of extensions of G by A.

$H^3(G,Center(G))$ = obstructions to existence of extensions of G by A.

2. Transfer and its applications: If $G$ is finite then

1) $H^i(G,M)$ is a torsion group annihilated by multiplication by $|G|$.

2) Embedding of $p$-primary component of $H^i(G,M)$ into a subgroup of $H^i(P,M)$, for any $p$-Sylow subgroup $P\subset G$.

3. In general, Brown's book "Cohomology of groups" gives a decent overview of what is good to know.

• In a sense this is just an interpretation of Josh's point (4) but +1 anyhow as it's an important interpretation. – Ryan Budney Jan 22 '10 at 5:17

Tate's theorem for the Tate cohomology (agrees with group cohomology for $r\geq1$) of finite groups states the following and is used heavily:

Let $G$ a finite group and $M$ a $G$-module and suppose that for all subgroups $H$ of $G$, $H^1_T(H,M)=0$ and $H^2_T(H,M)$ is cyclic of order $|H|$. Then for all $r$ there is an isomorphism $H^r_T(G,\mathbb{Z})\cong H^{r+2}_T(G,M)$.

See Milne's notes on 'Class field theory' or the wikipedia entry for example. This is commonly applied in the case of $G$ being a Galois group $Gal(L/K)$ and $M=L^\times$ for example.

• And more generally, the characterization of periodic groups as given, say, in Cartan-Eilenberg. – Mariano Suárez-Álvarez Jan 21 '10 at 16:19

Shapiro's lemma and dimension-shifting. Cohomology of cyclic groups and Herbrand quotients.

It would be helpful to know what you need to know group cohomology for.

If you have an interest in pro-p or profinite groups, there's a slew of things to add on here (notably, the interpretation of the ranks of $H^1(G,F_p)$ and $H^2(G,F_p)$ as cardinalities of minimal generator and relator sets, the value of the cup product and Massey products in determining the structure of these relations, the relation with the Schur Multiplicator given by Hopf's formula above). Similarly, if you're interested in group cohomology as Galois cohomology, there's a whole field of mathematics to add on to the list. In addition to class field theory via group cohomology a la Tate, there are a few papers (I remember a particularly good one by Cornell and Rosen) that derive a large portion of a semester of algebraic number theory starting from elementary group cohomology.

Here's one which is key for calculations: Let $H$ be a subgroup of $G$ and $W_G(H) = N_G(H)/H$. Then the restriction map $H^*(BG) \to H^*(BH)$ maps to the invariants $(H^*(BH))^{W_G(H)}$.

When $H$ is abelian, its cohomology is well-known (polynomial tensor exterior) and thus the cohomology of $G$ is mapping to something which can in principal be computed by invariant theory. Follow this with Quillen's theorem that the sum of these maps over all abelian subgroups has kernel which contains only nilpotent elements, and special cases such as Milgram's theorem that this is injective for symmetric groups, and you have a powerful computational tool.

Also, here is a nice survey by Alejandro Adem (whose book with Milgram is a good reference, complementary in many ways to Brown's). It is intended for a graduate student summer school audience: http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf

It will be pretty nice to learn some Galois cohomology and Class field theory, so that you can see these machineries put to good use.