Taut foliations of 3-manifolds with Heegaard genus 2 (Q6566409)

Let \(M\) be a closed orientable irreducible \(3\)-manifold. The L-space conjecture claims that the following three statements are equivalent: (1) \(M\) is not an L-space; (2) \(M\) admits a co-orientable taut foliation; (3) the fundamental group of \(M\) is left orderable. It is known that (2) implies (1), and that the conjecture holds for several classes of \(3\)-manifolds.\N\NRecently, Sarah Rasmussen showed that (3) implies (2) under the assumption that \(M\) has Heegaard genus two and some extra condition on the presentation of the fundamental group coming from a Heegaard diagram. The purpose of the present paper is to show the same conclusion without the latter condition. That is, the author proves that if a closed orientable irreducible \(3\)-manifold \(M\) has Heegaard genus two and the fundamental group is left orderable, then \(M\) admits a co-orientable taut foliation.\N\NThe argument is different from Rasmussen's construction. For a genus two Heegaard splitting, construct a \(2\)-complex from the Heegaard surface and complete sets of non-separating meridian disks of two handlebodies. Then this is deformed into a branched surface. For each branch sector, an element of the fundamental group is assigned so that no one is negative with respect to a fixed left ordering. By deleting branch sectors whose associated element is trivial, a new branched surface is obtained. It fully carries a lamination, which is, roughly speaking, extended to a co-orientable taut foliation. The last part needs a long analysis of branched surfaces.