  We construct a triangulation of a compactification of the Moduli space of a
surface with at least one puncture that is closely related to the
Deligne-Mumford compactification. Specifically, there is a surjective map from
the compactification we construct to the Deligne-Mumford compactification so
that the inverse image of each point is contractible. In particular our
compactification is homotopy equivalent to the Deligne-Mumford
compactification.
