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.



Rational Group Cohomology
Stallings showed that if $f:\Gamma \to \Delta$ is a homomorphism between finitely presented groups where $f\_i:H\_i(\Gamma,Q) \to H\_i(\Delta,Q)$ 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. 


