Automorphisms of the three-torus preserving a genus-three Heegaard splitting

Abstract

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. For the genus-three Heegaard splitting of the 3-torus, we find an eight element generating set for this group. Six of these generators induce generating elements of the mapping class group of the 3-torus and the remaining two are isotopy trivial in the 3-torus.

Bounding the stable genera of Heegaard splittings from below

“…The analogous lemma for isotopies of Morse functions is Lemma 9 in [7], and Lemma 7 can be proved by a similar argument. We allow the reader to work out the details.…”

On Critical Heegaard Splittings of Tunnel Number Two Composite Knot Exteriors

“…Note that each of D and E is the 1-handle connecting the neighborhood of two tori from the minus boundary of its compression body in the case (b) (consider the standard genus three splitting of T 3 and a weak reducing pair of this splitting.) In Lemma 6 of [5], Johnson proved that a weak reducing pair is determined uniquely by a compressing disk for the standard genus three splitting of T 3 and Theorem A.1 is the generalization of the Johnson's Lemma.…”

The mapping class groups of reducible Heegaard splittings of genus two

“…In other words, the theorem says that the mapping class groups of genus-2 Heegaard splittings of Hempel distance 0 are all finitely presented. It is shown in [20] and [16] that the mapping class groups are all finite for the Heegaard splittings of Hempel distance at least 4. The mapping class groups of the splittings of Hempel distances 2 and 3 still remain mysterious.…”