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\section*{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]].

\begin{itemize}%
\item [[David Mumford]], \emph{Towards an enumerative geometry of the moduli space of curves}, Arithmetic and geometry, Vol. II, Birkh\"a{}user Boston, Boston, MA, 1983, pp. 271--328, \href{http://www.ams.org/mathscinet-getitem?mr=85j:14046}{MR85j:14046}
\item E. Y. Miller, \emph{The homology of the mapping class group}, J. Differential Geom. \textbf{24} (1986), no. 1, 1--14.
\item John Harer, \emph{The cohomology of the moduli space of curves}, Lec. Notes in Math. 1337, p. 138--221. Springer, Berlin, 1988.
\item Ib Madsen, Michael Weiss, \emph{The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]}, Ann. of Math. (2) \textbf{165} (2007), no. 3, 843--941, \href{http://www.ams.org/mathscinet-getitem?mr=2009b:14051}{MR2009b:14051}, \href{http://dx.doi.org/10.4007/annals.2007.165.843}{doi}, \href{http://arxiv.org/abs/math.AT/0212321}{math.AT/0212321}
\item Shigeyuki Morita, \emph{Characteristic classes of surface bundles}, Invent. Math. \textbf{90} (1987), no. 3, 551--577, \href{http://dx.doi.org/10.1007/BF01389178}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=89e:57022}{MR89e:57022}
\item Sh. Morita, \emph{Introduction to mapping class groups of surfaces and related groups}, in: Handbook of Teichm\"u{}ller theory (A. Papadopoulos, editor), vol. I, EMS Publishing House, Z\"u{}rich, 2007, 353--386.
\item C. Faber, R. Pandharipande, \emph{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. \textbf{48} (2000), 215--252, \href{http://www.ams.org/mathscinet-getitem?mr=2002e:14041}{MR2002e:14041}, \href{http://dx.doi.org/10.1307/mmj/1030132716}{doi}
\item Carel Faber, \emph{Hodge integrals, tautological classes and Gromov-Witten theory}, Proc. Workshop \emph{Algebraic Geometry and Integrable Systems related to String Theory} (Kyoto, 2000). Srikaisekikenkysho Kkyroku \textbf{1232} (2001), 78--87, \href{http://www.ams.org/mathscinet-getitem?mr=1905884}{MR1905884}
\item K. Igusa, \emph{Graph cohomology and Kontsevich cycles}, Topology \textbf{43} (2004), n. 6, p. 1469-1510, \href{http://www.ams.org/mathscinet-getitem?mr=2005d:57028}{MR2005d:57028}, \href{http://dx.doi.org/10.1016/j.top.2004.03.004}{doi}
\item Gabriele Mondello, \emph{Riemann surfaces, ribbon graphs and combinatorial classes}, in: Handbook of Teichm\"u{}ller theory. Vol. II, 151--215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Z\"u{}rich, 2009; draft with index: \href{http://www.mat.uniroma1.it/~mondello/me/papers/ober-definitive.pdf}{pdf}, arxiv version \href{http://arxiv.org/abs/0705.1792}{math.AG/0705.1792}, \href{http://www.ams.org/mathscinet-getitem?mr=2010f:32012}{MR2010f:32012}
\item G. Mondello, \emph{Combinatorial classes on $\mathcal{M}_{g,n}$ are tautological}, Int. Math. Res. Not. \textbf{44} (2004), 2329---2390, \href{http://www.ams.org/mathscinet-getitem?mr=2005g:14056}{MR2005g:14056}, \href{http://dx.doi.org/10.1155/S1073792804131462}{doi}, \href{http://arxiv.org/abs/math/0303207}{math.AG/0303207}

\end{itemize}
[[!redirects Mumford classes]] [[!redirects Mumford's class]] [[!redirects Mumford's classes]] [[!redirects Mumford-Morita-Miller classes]]



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