Cohomology of nilmanifolds and torsion-free, nilpotent groups

Abstract:

Let $M$ be a nilmanifold, i.e. $M = G/D$ where $G$ is a simply connected, nilpotent Lie group and $D$ is a discrete uniform, nilpotent subgroup. Then $M \simeq K(D,1)$. Now $D$ has the structure of an algebraic group and so has an associated algebraic group Lie algebra $L(D)$. The integral cohomology of $M$ is shown to be isomorphic to the Lie algebra cohomology of $L(D)$ except for some small primes depending on $D$. This gives an effective procedure for computing the cohomology of $M$ and therefore the group cohomology of $D$. The proof uses a version of form cohomology defined for subrings of ${\mathbf {Q}}$ and a type of Hirsch Lemma. Examples, including the important unipotent case, are also discussed.