  We study $b'_1(M)$, the co-rank of the fundamental group of a smooth closed
connected manifold $M$. We calculate this value for the direct product of
manifolds. We characterize the set of all possible combinations of $b'_1(M)$
and the first Betti number $b_1(M)$ by explicitly constructing manifolds with
any possible combination of $b'_1(M)$ and $b_1(M)$ in any given dimension.
Finally, we apply our results to the topology of Morse form foliations. In
particular, we construct a manifold $M$ and a Morse form $\omega$ on it for any
possible combination of $b'_1(M)$, $b_1(M)$, $m(\omega)$, and $c(\omega)$,
where $m(\omega)$ is the number of minimal components and $c(\omega)$ is the
maximum number of homologically independent compact leaves of $\omega$.
