  We define the surface complex for $3$-manifolds and embark on a case study in
the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered
space captures some of the topology of the Seifert fibered space, so, not
surprisingly, the surface complex of a Seifert fibered space always contains a
subcomplex isomorphic to the curve complex of the base orbifold.
