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

TL;DR

This paper investigates the co-rank of the fundamental group of manifolds, computes it for product manifolds, characterizes possible value combinations with Betti numbers, and applies findings to Morse form foliations.

Contribution

It explicitly constructs manifolds with any combination of co-rank and Betti number, and applies these results to the topology of Morse form foliations.

Findings

Calculated co-rank for product manifolds.

Characterized all possible co-rank and Betti number combinations.

Constructed manifolds and Morse forms for all possible parameter combinations.

Abstract

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 $\omega$.