Rational Group Cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:12:01Z http://mathoverflow.net/feeds/question/5057 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5057/rational-group-cohomology Rational Group Cohomology James Griffin 2009-11-11T15:12:38Z 2012-02-16T09:07:57Z <p>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:</p> <p>1) Are there any favoured examples that you would recommend a look at? (Recommended references would be just as welcome.)</p> <p>And the main question:</p> <p>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?</p> <p>I have a feeling that someone with a good knowledge of rational homotopy theory would be able to answer this question with relative ease.</p> http://mathoverflow.net/questions/5057/rational-group-cohomology/5064#5064 Answer by Danny Calegari for Rational Group Cohomology Danny Calegari 2009-11-11T15:58:00Z 2009-11-11T15:58:00Z <p>Stallings showed that if $f:\Gamma \to \Delta$ is a homomorphism between finitely presented groups where <img src="http://latex.mathoverflow.net/png?%24f%5Fi%3AH%5Fi%28%5CGamma%2CQ%29%20%5Cto%20H%5Fi%28%5CDelta%2CQ%29%24" alt="$f\sb i:H\sb i(\Gamma,Q) \to H\sb i(\Delta,Q)$" title="" /> 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 . . .</p> <p>Stallings' paper is </p> <p>MR0175956 (31 #232) Stallings, John Homology and central series of groups. J. Algebra 2 1965 170--181. </p> http://mathoverflow.net/questions/5057/rational-group-cohomology/88613#88613 Answer by Mark Grant for Rational Group Cohomology Mark Grant 2012-02-16T09:07:57Z 2012-02-16T09:07:57Z <p>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.</p> <p>A good reference (with a crash course in rational homotopy theory included) is the paper</p> <p>Oprea, John <em>The category of nilmanifolds.</em> Enseign. Math. (2) 38 (1992), no. 1-2, 27–40.</p>