On the Betti Numbers of Nilpotent Lie Algebras of Small Dimension

Abstract

The work of Golod and Šafarevič on class field towers motivated the conjecture that b2 > b2 1/4 for nilpotent Lie algebras of dimension at least 3, where b i denotes the i th Betti number. Using a new lower bound for b 2 and a characterization of Lie algebras of the form g/Z(g), we prove this conjecture for 2-step algebras. We also give the Betti numbers of nilpotent Lie algebras of dimension at most 7 and use them to establish the conjecture for all nilpotent Lie algebras whose centres have codimension ≤ 7.KeywordsBetti NumberRiemannian FoliationDense LeafAbelian IdealMaximal Abelian SubalgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.