  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 $3$-manifold or a spherical
$3$-manifold with large fundamental group satisfies the condition if it is
topologically minimal in the sense of Bachman. In conclusion, such a Heegaard
splitting has the finitely generated mapping class group.
