  We give examples of closed hyperbolic 3-manifolds with first Betti number 2
and 3 for which no sequence of finite abelian covering spaces increases the
first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a
characterization in terms of some generalized self-linking numbers of $M$, for
there to exist a family of $\mathbb{Z}_n$ covering spaces,
  $M_n$, in which $\beta_1(M_n)$ increases linearly with $n$.
  The latter generalizes work of M. Katz and C. Lescop [KL], by showing that
the non-vanishing of any one of these invariants of $M$ is sufficient to
guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and
Katz [IK]).
