nLab
Mumford class

There is a very beautiful story on several families of stable classes of the compactification g,n\mathcal{M}_{g,n} of the moduli space of algebraic genus gg curves with nn 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 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 A_\infty-algebras; Hamilton-Lazarev have shown that the recipe for the classes from the A 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 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)