The co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliations

We study $b_1^{\mathrm{\prime }}(M)$, 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 $b_1^{\mathrm{\prime }}(M)$ and the first Betti number $b_1(M)$ by explicitly constructing manifolds with any possible combination of $b_1^{\mathrm{\prime }}(M)$ 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 $b_1^{\mathrm{\prime }}(M)$, $b_1(M)$, $m(\omega )$, 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 $\omega$.