nLab Mumford class

There is a very beautiful story on several families of stable classes of the compactification $\mathcal{M}_{g,n}$ of the moduli space of algebraic genus $g$ curves with $n$ marked points, playing major role in geometry (especially intersection theory), Gromov-Witten theory, etc. They are sometimes referred by standard notation and sometimes by their names; one has families of Mumford-Miller classes and of Morita classes.

Kontsevich associates to any cyclic/symplectic $A_\infty$-algebra certain partition function which is a inhomogeneous class in graph homology. Igusa and Mondello have substantiated Kontsevich’s claim that the Mumford-Miller-Morita classes are induced from certain family of $A_\infty$-algebras;

Related entries Witten conjecture, mapping class group.

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• K. Igusa, Graph cohomology and Kontsevich cycles, Topology 43 (2004), n. 6, p. 1469-1510, MR2005d:57028, doi
• Gabriele Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, in: Handbook of Teichmüller theory. Vol. II, 151–215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009; draft with index: pdf, arxiv version math.AG/0705.1792, MR2010f:32012
• G. Mondello, Combinatorial classes on $\mathcal{M}_{g,n}$ are tautological, Int. Math. Res. Not. 44 (2004), 2329-–2390, MR2005g:14056, doi, math.AG/0303207