Thick isotopy property and the mapping class groups of Heegaard splittings

TL;DR

This paper establishes a criterion linking the fundamental group of the space of Heegaard splittings to the thick isotopy property, showing that certain minimal splittings have finitely generated mapping class groups.

Contribution

It provides a necessary and sufficient condition for the fundamental group of Heegaard splitting spaces to be finitely generated, connecting isotopy properties with topological minimality.

Findings

Heegaard splitting spaces have finitely generated fundamental groups under the thick isotopy condition.

Topologically minimal splittings with finitely generated disk complex homotopy groups satisfy the condition.

Such splittings have finitely generated mapping class groups.

Abstract

We give a necessary and sufficient condition for the fundamental group of the space of Heegaard splittings of an irreducible $3$-manifold to be finitely generated. The condition is exactly the conclusion of the thick isotopy lemma proved by Colding, Gabai and Ketover, which says that any isotopy of a Heegaard surface is achieved by a $1$-parameter family of surfaces with area bounded above by a universal constant and with some ``thickness property''. We also prove that a Heegaard splitting of a hyperbolic or spherical $3$-manifold satisfies the condition if it is topologically minimal (in the sense of Bachman) and its disk complex has finitely generated homotopy group. In conclusion, such a Heegaard splitting has finitely generated mapping class group.