The Heegaard Genus of Bundles over S

This is a largely expository paper exploring theorems of Rubinstein and Lackenby. Rubinstein’s Theorem studies the Heegaard genus of certain hyperbolic 3-manifolds that fiber over S and Lackenby’s Theorem studies the Heegaard genus of certain Haken manifolds. Our target audience are 3-manifold theorists with good understanding of Heegaard splittings but perhaps little experience with minimal surfaces. The main purpose of this note is generalizing Rubinstein’s minimal surface argument (Section 4) and explaining the minimal surface technology necessary for that theorem (Section 3). We assume familiarity with the basic notions of 3-manifold theory (e.g [4][6]), the basic nations about Heegaard splittings (e.g. [13]), and Casson– Gordon’s [1] concept of strong irreducibility/weak reducibility. In Section 5 we assume familiarity with Scharlemann–Thompson untelescoping. All manifolds considered in this paper are closed, orientable 3-manifolds and all surfaces considered are closed and orientable. By the genus of a 3-manifold M (denoted g(M)) we mean the genus of a minimal genus Heegaard surface for M . In [12] Rubinstein used minimal surfaces to study the Heegaard genus of hyperbolic manifolds that fiber over S, more precisely, of closed 3manifolds (say Mφ or simply M when there is no place for confusion) that fiber over the circle with fiber a closed surface of genus g and pseudoAnosov monodromy (say φ). While there exist genus two manifolds that fiber over S with fiber of arbitrarily high genus (for example, consider 0surgery on 2 bridge knots with fibered exterior [3]) Rubinstein showed that this is often not the case: a manifold that fibers over S with genus g fiber has a Heegaard surface of genus 2g + 1 that is obtained by taking two disjoint fibers and tubing them together once on each side. We call this surface and surfaces obtained by stabilizing it standard. M has a cyclic cover of degree d (denoted Mφd or simply Md), dual to the fiber, whose monodromy is φ. Rubinstein shows that for small h and large d any Heegaard surface for Md of genus ≤ h is a stabilization of the standard Heegaard surface of