  A classical combinatorial fact is that the simplicial complex consisting of
disjointly embedded chords in a convex planar polygon is a sphere. For any
surface F with non-empty boundary, there is an analogous complex Arc(F)
consisting of suitable equivalence classes of arcs in F connecting its boundary
components. The main result of this paper is the determination of those arc
complexes Arc(F) that are also spherical. This classification has consequences
for Riemann's moduli space via its known identification with an analogous arc
complex in the punctured case with no boundary. Namely, the essential
singulities of the natural cellular compactification can be described.
