As is well known, there is a combinatorial model for Riemann’s moduli space M of a punctured surface F given by the geometric realization of the partially ordered set of all appropriate arc families filling F. It is natural to consider the combinatorial compactification of M provided by all appropriate arc families in F, where we drop the restriction that the arc families fill. We had conjectured almost 20 years ago that a version of this combinatorial compactification is an orbifold, and this in turn was shown to follow from a related conjecture that the analogous arc complexes for surfaces with boundary are spherical. We have recently shown with Dennis Sullivan that these conjectures are FALSE in general and have given a complete list of those surfaces for which these conjectures do hold. The context and background of these conjectures will also be discussed.

University of Southern California, Los Angeles