Question

This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into two:

1) Are there any favoured examples that you would recommend a look at? (Recommended references would be just as welcome.)

And the main question:

2) What sort of functors on the category of groups leave the rational cohomology unchanged? In particular is there a projection onto a special subcategory of groups that is in some way the right category to study?

I have a feeling that someone with a good knowledge of rational homotopy theory would be able to answer this question with relative ease.

Answer 1

Stallings showed that if $f:\Gamma \to \Delta$ is a homomorphism between finitely presented groups where is bijective for $i\le 1$ and surjective for $i=2$ then $f\otimes Q: \Gamma \otimes Q \to \Delta \otimes Q$ is an isomorphism between the $Q$-unipotent completions. So "taking the $Q$-unipotent completion" is a functor with some interesting properties with respect to rational group (co)homology. I'm not sure if this is the kind of thing you're after . . .

Stallings' paper is

MR0175956 (31 #232) Stallings, John Homology and central series of groups. J. Algebra 2 1965 170--181.

Answer 2

One class of groups whose rational cohomology is particularly easy to handle are the finitely generated torsion-free nilpotent groups. This is because they admit refined Postnikov systems which can be localised, leading to a recipe for computing the Sullivan minimal model of cochains in terms of their central series. This means one can also read off cup and Massey products quite easily.

A good reference (with a crash course in rational homotopy theory included) is the paper

Oprea, John The category of nilmanifolds. Enseign. Math. (2) 38 (1992), no. 1-2, 27–40.