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.

I picked the isotopy subgroup rather than the whole mapping class group because the mapping class group may have finite order elements coming from automorphisms of the ambient 3-manifold. On the other hand, Hyam and I show that if is hyperbolic then the isotopy subgroup is torsion free, so we don’t need to worry about finite order elements.

Constructing irreducible Heegaard splittings with reducible automorphisms turns out not to be too difficult once you’ve had some practice. For example, the union of any two pages of an open book decomposition forms a Heegaard surface, and we can “spin” this Heegaard surface around the monodromy of this open book. The fixed set of this automorphism is the binding of the open book, which is separating in the Heegaard surface, and the restriction to each of the complementary components is equal to the monodromy map for the open book. These automorphisms form a (usually infinite) cyclic subgroup, and I recently showed that for many open book decompositions, this is the entire isotopy subgroup of the induced Heegaard surface [3].

A second, more general, construction is to find a handle on one side of the Heegaard surface that has some flexibility. For example if the Heegaard surface has a pair of disks on opposite sides that intersect in exactly two points, then a regular neighborhood of the two disks is a solid torus. The Heegaard surface intersects the boundary of this solid torus in two or four points depending on the signs of the two intersections between the disks. You can take the handle dual to one of these disks and pull it around the longitude of the solid torus, inducing Dehn twists along the two or four loops. Pairs of disks like this are quite common and the set of automorphisms defined in this way generate the isotopy subgroup for the (unique) genus three Heegaard splitting of the 3-torus [4].

You can find more interesting subgroups by generalizing this construction. For example, if is a one-sided surface, then a regular neighborhood of is a twisted interval bundle. If you attach a tube to along one of the vertical intervals, the resulting surface is a Heegaard surface. (This is left as an exercise.) Moreover, there is an automorphism of this Heegaard surface defined by draging this tube along any path in the one-sided surface and back to itself. Thus the isotopy subgroup of the Heegaard surface has a subgroup isomorphic to the fundamental group of the one-sided surface. (I’m currently finishing up a preprint showing that for many one-sided Heegaard surfaces, this is the entire isotopy subgroup of the induced two-sided Heegaard surface.)

There are a few other ways to generalize this tube dragging construction as well, which I won’t write about here. I haven’t figured out a way to generalize the construction to more than one handle at a time, but I think it would be very interesting to find examples of automorphisms defined by isotoping a larger portion of the surface, i.e. more than just a single handle, around inside of .

[2] Long, D. D., On pseudo-Anosov maps which extend over two handlebodies. *Proc. Edinburgh Math. Soc. (2)* 33 (1990), no. 2, 181–190.

## Leave a Reply