# 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; Hamilton-Lazarev have shown that the recipe for the classes from the $A_\infty$-algebra is a homotopy invariant and that the partition function takes values in the relative homology of an infinite-dimensional algebra of noncommutative Hamiltonians. They interpret the class as being somewhat analogous to Chern classes and call it a characteristic class of the $A_\infty$-algebra.

Related entries Witten conjecture, mapping class group.

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Revised on September 9, 2010 21:20:09 by Zoran Škoda (161.53.130.104)