Affine Hirsch foliations on 3-manifolds

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. Then, we analysis the 3-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). On 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 ∈ 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.