(Reposted from my old ldt blog)

Now, those of you who have been unfortunate enough to see me give a talk in the last year know that I have been thinking a lot about mapping class groups of Heegaard splittings. These are groups of automorphisms of a 3-manifold that take a Heegaard surface onto itself, modulo isotopies of the 3-manifold that preserve the Heegaard surface setwise. When I was visiting Rutgers, New Brunswick a few months ago, spreading the gospel of the mapping class group, Saul Schleimer suggested an interesting idea and I hope he will forgive me for mentioning it here.Any automorphism of a Heegaard splitting that preserves each handlebody induces an automorphism the fundamental group of each handlebody (which is a free group). Saul suggested trying to understand this pair of actions of the mapping class group on these two free groups. If you Dehn twist the boundary of a handlebody along the boundary of a properly embedded disk, the action on the fundamental group is trivial. Luft [1] showed, moreover, that any automorphism of a handlebody that acts trivially on the fundamental group is a composition of Dehn twists along properly embedded disks.

If a Heegaard splitting is reducible, i.e. there is an embedded sphere that intersects the Heegaard surface in a single essential loop, then Dehn twisting along this loop extends into each handlebody as a Dehn twist along a properly embedded disk. Thus such an automorphism acts trivially on the fundamental groups of both handlebodies. The question is whether there is a converse analogous to Luft’s result. In other words, given an automorphism of a Heegaard splitting that fixes the fundamental groups of both handlebodies, is this automorphism always a composition of twists along reducing spheres? A slightly weaker question is: Is there an irreducible Heegaard splitting that admits an automorphism that fixes both fundamental groups?

In the case when the the automorphism is a composition of disjoint Dehn twists, a positive answer to the first question follows from a result proved by Oertel [2] and and generalized by McCullough [3]. Oertel’s theorem states that if a composition of Dehn twists along disjoint loops extends into a handlebody then these loops cobound properly embedded disks and annuli in the handlebody. If an automorphism of a Heegaard surface is a composition of Dehn twists along disjoint loops then the disks and annuli in the two handlebodies form spheres and tori in the ambient manifold. If the automorphism fixes the fundamental groups of both handlebodies then these surfaces must in fact be reducing spheres.

Note also that if one can find a sphere that intersects one handlebody in a pair of non-parallel essential disks and the other handlebody in an annulus then Dehn twisting the Heegaard surface along the loops on intersection induces an automorphism that fixes the fundamental group of one handlebody but not the other. It should be possible to construct an irreducible (but weakly reducible) Heegaard splitting that has such a sphere.

[1] Actions of the homeotopy group of an orientable $3$-dimensional handlebody. Math. Ann. 234 (1978), no. 3, 279–292.

[2] U. Oertel. Automorphisms of 3-dimensional handlebodies. Topology, 41:363–410, 2002.

[3] Follow link to arXiv.

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