# Rational Group Cohomology

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.

• DC's answer is very good and I'm going to accept it, but I don't yet understand why there might not be a stronger result. I would welcome further answers which could enlighten me and the rest of MathOverflow. – James Griffin Nov 18 '09 at 9:45

## 2 Answers

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.

• Thank you, a "Q-unipotent completion" sounds likely; I shall get hold of that paper. Would I be right or wrong in guessing that a homomorphism inducing an isomorphism between Q-unipotent completions induces an isomorphism between the homology groups? – James Griffin Nov 11 '09 at 16:39
• I assume you mean "induces an isomorphism between homology in dimension 1 and a surjection in dimension 2". Off the top of my head, I don't see why this is true (or false) but maybe it is easy to see. – Danny Calegari Nov 11 '09 at 17:20
• Well I've had a look at the paper and it looks like it probably pre-dates the notion (at least in name) of Q-unipotent completion. Can you recommend a reference? I not had any luck on Google. I think that your correction to my question in the comment above is a very sensible one. – James Griffin Nov 11 '09 at 17:29
• I'm definitely no expert, so unfortunately I don't know a canonical reference. One reference would be Quillen's "Rational homotopy theory" from 1969 (maybe that doesn't use the name "Q-unipotent completion" either?) or Appendix A to "Fundamental groups of compact Kahler manifolds" by Amoros et. al. (AMS Math. Surveys and Monographs vol. 44). Sorry not to be more help. – Danny Calegari Nov 11 '09 at 17:47

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.

• Thank you Mark, interestingly enough I've been reading recently about the homology of the group of upper uni-triangular matrices with integer coefficients. – James Griffin Feb 17 '12 at 9:43