  We show that if a Heegaard splitting is obtained by gluing a splitting of
Hempel distance at least 4 and the genus-1 splitting of $S^2 \times S^1$, then
the Goeritz group of the splitting is finitely generated. To show this, we
first provide a sufficient condition for a full subcomplex of the arc complex
for a compact orientable surface to be contractible, which generalizes the
result by Hatcher that the arc complexes are contractible. We then construct
infinitely many Heegaard splittings, including the above-mentioned Heegaard
splitting, for which suitably defined complexes of Haken spheres are
contractible.
