Recall that the *mapping class group* of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo isotopies of that keep on itself. The *isotopy subgroup* is the group of such maps that are isotopy trivial on , when you ignore the Heegaard surface. Hyam Rubinstein and I constructed a number of examples of Heegaard splittings with non-trivial isotopy subgroups [1], but all of these groups were generated by reducible automorphisms. Before our paper, Darren Long had constructed a strongly irreducible Heegaard splitting with a pseudo-Anosov element in its mapping class group [2] but the ambient manifold is a Seifert fibered space over a sphere with five singular fibers and Heegaard splittings of Seifert fibered spaces have large mapping class groups, usually including lots of reducible maps. This raises the question: Is there a Heegaard splitting with a non-trivial isotopy subgroup consisting entirely of pseudo-Anosov maps (other than the identity)? I will go a step further and, in the spirit of my last two posts, suggest a reckless conjecture:

**The reducible automorphism conjecture:** The isotopy subgroup of every Heegaard splitting is generated by reducible automorphisms.