nLab
Mumford class

There is a very beautiful story on several families of stable classes of the compactification ${ℳ}_{g,n}$ 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 $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-algebras; Hamilton-Lazarev have shown that the recipe for the classes from the $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-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 ${ℳ}_{g,n}$ 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.