  We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with
${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be
regarded as a qualitative improvement of a result due to Culler and Shalen,
because the constant $3.69$ is greater than the ordinal corresponding to
$\omega^2$ in the well-ordered set of finite volumes of hyperbolic
$3$-manifolds. We also show that if ${\rm vol}\ M\le 3.77$, we have ${\rm dim}\
H_1(M;{\bf F}_2)\le10$.
  These results are applications of a new method for obtaining lower bounds for
the volume of a closed, orientable hyperbolic $3$-manifold such that $\pi_1(M)$
is $k$-free for a given $k\ge4$. Among other applications we show that if
$\pi_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound
of $3.44$ given by Culler and Shalen), and that if $\pi_1(M)$ is $5$-free we
have ${\rm vol}\ M>3.77$.
