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

Abstract

Abstract We study b 1 ′ $b_{1}'$ (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 ′ $b_{1}'$ (M) and the first Betti number b 1(M) by explicitly constructing manifolds with any possible combination of b 1 ′ $b_{1}'$ (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 ω on it for any possible combination of b 1 ′ $b_{1}'$ (M), b 1(M), m(ω), and c(ω), where m(ω) is the number of minimal components and c(ω) is the maximum number of homologically independent compact leaves of ω.