Abstract
A well-known 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 QA(F) consisting of equivalence classes of arcs in F connecting a given finite set of points in its boundary modulo diffeomorphisms of F pointwise fixing the boundary and any punctures. The main result of this paper is the determination of those complexes QA(F) that are also spheres. This classification has consequences for Riemann's moduli space of curves via its known identification with a related quotient arc complex in the punctured case with no boundary. Namely, the essential singularities of the natural cellular compactification of Riemann's moduli space can be described.
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