Betti number estimates for nilpotent groups
Michael Freedman, Richard Hain, and Peter Teichner

Abstract. We prove an extension of the following result of Luboszky and
Magid on the rational cohomology of a nilpotent group $G$: if $b_1 <
\infty$ and $G \otimes \mathbb Q \ne 0, \mathbb Q, \mathbb Q^2$ then
$b_2 > b_1^2/4$. Here the $b_i$ are the rational Betti numbers of $G$
and $G \otimes \mathbb Q$ denotes the Malcev-completion of $G$. In the
extension, the bound is improved when we know that all relations of $G$
all have at least a certain commutator length. As an application of the
refined inequality, we show that each closed oriented $3$-manifold falls
into exactly one of the following classes: it is a rational homology
$3$-sphere, or it is a rational homology $S^1 \times S^2$, or it has 
the rational homology of one of the oriented circle bundles over the 
torus (which are indexed by an Euler number $n \in \mathbb Z$, e.g.
$n = 0$ corresponds to the $#$-torus) or it is of {\em general type\/}
by which we mean that the rational lower central series of the 
fundamental group does not stabilize. In particular, any $3$-manifold
group which allows a maximal torsion-free nilpotent quotient admits
a rational homology isomorphism to a torsion-free nilpotent group.