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.
Categories: math.GT