Affine Hirsch foliations on 3-manifolds

Abstract

This paper is devoted to discussing affine Hirsch foliations on $3$-manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations. Furthermore, every case is possible. Then, we analyze the $3$-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (abbreviated as DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.