Math 541 : The Mumford Conjecture
Syllabus : [pdf]
I will use this page to record references for various things.
Mumford's conjecture : classical papers.
Mumford's conjecture was first stated in the following paper. Unfortunately,
I do not have an electronic copy of it.
MR0717614 (85j:14046)
Mumford, David(1HRV)
Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry, Vol. II, 271328,
Progr. Math., 36, BirkhĂ¤user Boston, Boston, MA, 1983.
For topologists, it is probably easier to read the following two papers, which give topological constructions
of the MMM (=Miller Morita Mumford) classes and show that they generate a polynomial algebra in the stable homology
of the mapping class group.
[pdf] Miller, E. "The homology of the mapping class group", J. Differential Geom. 24 (1986), no. 1, 114.
[pdf] Morita, S. "Characteristic classes of surface bundles", Invent. Math. 90 (1987), no. 3, 551577.
Instead of Morita's paper, you might want to read his book. Unfortunately, I do not have an electronic copy of it.
MR1826571 (2002d:57019)
Morita, Shigeyuki
Geometry of characteristic classes.
Translated from the 1999 Japanese original. Translations of Mathematical Monographs, 199. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001. xiv+185 pp. ISBN: 0821821393
The proof of the Mumford conjecture.
The Mumford conjecture was first proven in the following landmark paper of Madsen and Weiss.
[pdf] Madsen, I and Weiss, M, "The stable moduli space of Riemann surfaces: Mumford's conjecture",
Ann. of Math. (2) 165 (2007), 843941.
This paper actually proves a stronger result which was first conjectured to hold in the following paper.
[pdf] Madsen, I and Tillmann, U, "The stable mapping class group and Q(CP^{\infty}_+)", Invent. Math. 145 (2001), 509544.
There have been a number of surveys about the original proof and related ideas. I recommend the following ones.
[pdf] I. Madsen, Moduli spaces from a topological viewpoint, International Congress of Mathematicians. Vol. I, 385411, Eur. Math. Soc., Zurich, 2007.
[pdf] I. Madsen and M. Weiss, The stable mapping class group and stable homotopy theory. European Congress of Mathematics, 283307, Eur. Math. Soc., Zurich, 2005.
[pdf] G. Powell, The Mumford conjecture (after Madsen and Weiss), Seminaire Bourbaki. Vol. 2004/2005. Asterisque No. 307 (2006), Exp. No. 944, viii, 247282.
[pdf] U. Tillmann, Mumford's conjecture  a topological outlook, to appear in the Handbook of Moduli Spaces.
[pdf] M. Weiss, Cohomology of the stable mapping class group, in Topology, Geometry and Quantum Field Theory, 379404, Lond. Math. Soc. Lecture Note Ser., 308, Cambridge University Press, 2004.
Alternate proofs of the Mumford conjecture.
There are now a number of alternate proofs of the Mumford conjecture. Here are a few.
[pdf] S. Galatius, I. Madsen, U. Tillmann, M. Weiss: The Homotopy Type of the Cobordism Category. Acta Math. 202, no. 2, (2009) 195239.
[pdf] Y. Eliashberg, S. Galatius, and N. Mishachev. MadsenWeiss for geometrically minded topologists, Geometry & Topology 15 (2011) 411472.
[pdf] S. Galatius and O. RandalWilliams. Monoids of moduli spaces of manifolds, Geometry & Topology 14 (2010) 12431302.
We will be covering this last proof in this class. Unfortunately, the paper by Galatius and RandalWilliams proves a much more general
result and is thus rather hard to read. Thankfully, there are now two nice surveys of this new proof.
[pdf] A. Hatcher, A short exposition of the MadsenWeiss theorem.
[pdf] S. Galatius, Lectures on the MadsenWeiss theorem.
There is also a nice (but incomplete) set of lecture notes by Tom Church available here.
