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.

- David Mumford,
*Towards an enumerative geometry of the moduli space of curves*, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, MR85j:14046 - E. Y. Miller,
*The homology of the mapping class group*, J. Differential Geom.**24**(1986), no. 1, 1–14. - John Harer,
*The cohomology of the moduli space of curves*, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988. - Ib Madsen, Michael Weiss,
*The stable moduli space of Riemann surfaces: Mumford's conjecture*, Ann. of Math. (2)**165**(2007), no. 3, 843–941, MR2009b:14051, doi, math.AT/0212321 - Shigeyuki Morita,
*Characteristic classes of surface bundles*, Invent. Math.**90**(1987), no. 3, 551–577, doi, MR89e:57022 - Sh. Morita,
*Introduction to mapping class groups of surfaces and related groups*, in: Handbook of Teichmüller theory (A. Papadopoulos, editor), vol. I, EMS Publishing House, Zürich, 2007, 353–386. - C. Faber, R. Pandharipande,
*Logarithmic series and Hodge integrals in the tautological ring*, With an appendix by Don Zagier. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J.**48**(2000), 215–252, MR2002e:14041, doi - Carel Faber,
*Hodge integrals, tautological classes and Gromov-Witten theory*, Proc. Workshop*Algebraic Geometry and Integrable Systems related to String Theory*(Kyoto, 2000). Sūrikaisekikenkyūsho Kōkyūroku**1232**(2001), 78–87, MR1905884 - 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 - Alastair Hamilton, Andrey Lazarev,
*Characteristic classes of A-infinity algebras*, J. Homotopy Relat. Struct.**3**(2008), no. 1, 65–111, math.QA/0608395 - Alastair Hamilton,
*Classes on compactifications of the moduli space of curves through solutions to the quantum master equation*, Lett. Math. Phys.**89**(2009), no. 2, 115–130.

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