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Moduli spaces of curves (with nonempty boundary or at least one marked point) admit cell decompositions in which the cells are labelled by ribbon graphs. In fact, the moduli space of normalised metric ribbon graphs with $n$ boundary components is homeomorphic to the product of the moduli space of curves and an $(n-1)$-simplex.

From this geometric statement one can show that the ribbon graph complex (a chain complex spanned by ribbon graphs, with the differential given by summing over all edge contractions) computes the cohomology of moduli spaces of curves.

My question is: does anyone know how to describe the cochain level cup product structure on the ribbon graph complex? By general machinery, if a complex computes the cohomology of a space then it carries an $A_\infty$ structure for which it is equivalent to the cochains on the space with their usual $A_\infty$ structure.

I would like to know if there is a way to write down a combinatorial formula for this $A_\infty$ structure on the ribbon graph complex.

Similarly, the Lie graph complex (in which vertices are labelled by words in a generic Lie algebra) computes the cohomology of the spaces $BOut(F_n)$ (the classifying spaces of outer automorphism groups of free groups). Is there a way to describe the resulting $A_\infty$ structure on the Lie graph complex?

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Are moduli spaces of curves (rationally) formal? DM space should be formal, right? – Kevin H. Lin Apr 20 '10 at 10:23
Rational formality for moduli of smooth curves is not known, although Voronov showed roughly that they are formal in the Harer stable range. The DM compactifications, being compact Kahler orbifolds are certainly formal, and in fact, there is a nice paper where the modular operad they form is shown to be formal. – Jeffrey Giansiracusa Apr 21 '10 at 7:17

There is a way to obtain these kinds of spaces as geometric realizations of categories of graphs. See Igusa's book on Reidemeister torison. This gives a simplicial model. Basically its the barycentric subdivision. There is a standard formula for the cup product in this context which will probably work.

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yes, one can write down cup products by choosing a simplicial approximation of the diagonal, but then for the full A-infinity structure one needs some choices of explicit homotopies and higher homotopies to make the diagonal coassociative. A brute force approach becomes pretty much impossible and I was wondering if someone might have managed to do something more clever. – Jeffrey Giansiracusa Apr 21 '10 at 19:11

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