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(1-HRV)
    Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry, Vol. II, 271-328,
    Progr. Math., 36, Birkhäuser 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, 1-14.
  • [pdf] Morita, S. "Characteristic classes of surface bundles", Invent. Math. 90 (1987), no. 3, 551-577.

    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: 0-8218-2139-3

    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), 843-941.

    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), 509-544.

    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, 385-411, Eur. Math. Soc., Zurich, 2007.
  • [pdf] I. Madsen and M. Weiss, The stable mapping class group and stable homotopy theory. European Congress of Mathematics, 283-307, 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, 247-282.
  • [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, 379-404, 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) 195-239.
  • [pdf] Y. Eliashberg, S. Galatius, and N. Mishachev. Madsen-Weiss for geometrically minded topologists, Geometry & Topology 15 (2011) 411-472.
  • [pdf] S. Galatius and O. Randal-Williams. Monoids of moduli spaces of manifolds, Geometry & Topology 14 (2010) 1243-1302.
    We will be covering this last proof in this class. Unfortunately, the paper by Galatius and Randal-Williams 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 Madsen-Weiss theorem.
  • [pdf] S. Galatius, Lectures on the Madsen-Weiss theorem.
    There is also a nice (but incomplete) set of lecture notes by Tom Church available here.