Surface bundles with genus two Heegaard splittings

Abstract

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, with three exceptional fibers, and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with genus two Heegaard splittings and show that each is the result of integral Dehn surgery in one of these Seifert fibered spaces along loops where the Heegaard surface intersects a horizontal surface. (This type of surgery preserves both the bundle structure and the Heegaard splitting.)