On the Betti Numbers of Nilpotent Lie Algebras of Small Dimension

The work of Golod and Šafarevič on class field towers motivated the conjecture that b₂ > b2₁/4 for nilpotent Lie algebras of dimension at least 3, where b _i denotes the ith Betti number. Using a new lower bound for b₂ 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.