  Let $M=H_1\cup_S H_2$ be a Heegaard splitting of a closed orientable
3-manifold $M$ (or a bridge decomposition of a link exterior). Consider the
subgroup $\mathrm{MCG}^0(H_j)$ of the mapping class group of $H_j$ consisting
of mapping classes represented by auto-homeomorphisms of $H_j$ homotopic to the
identity, and let $G_j$ be the subgroup of the automorphism group of the curve
complex $\mathcal{CC}(S)$ obtained as the image of $\mathrm{MCG}^0(H_j)$. Then
the group $G=<G_1, G_2>$ generated by $G_1$ and $G_2$ preserve the homotopy
class in $M$ of simple loops on $S$. In this paper, we study the structure of
the group $G$ and the problem to what extent the converse to this observation
holds.
