Nilmanifolds and associated Lie algebras over the integers

Abstract

We consider the correspondence between nilmanifolds and Lie algebras with rational basis, and we dene spectral sequences converging to the respective cohomologies. The E2 terms of the spectral sequences are the cohomolgies of isomorphic graded Lie algebras. Each nilmanifold gives rise to a Lie algebra with rational basis. We give an example which illustrates that not all such Lie algebras correspond to nilmanifolds. Given a Lie algebra with rational basis we give a construction that produces a nilmanifold with Lie algebra that is rationally equivalent to the starting Lie algebra.