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We construct a triangulation of a compactification of the Moduli space of asurface with at least one puncture that is closely related to theDeligne-Mumford compactification. Specifically, there is a surjective map fromthe compactification we construct to the Deligne-Mumford compactification sothat the inverse image of each point is contractible. In particular ourcompactification is homotopy equivalent to the Deligne-Mumfordcompactification.

A triangulation of a homotopy-Deligne-Mumford compactification of the Moduli of curves