nLab Mumford's conjecture

Idea

A conjecture in (Mumford 83) concerned the ordinary cohomology with rational numbers coefficients of the classifying space of the stable mapping class group, which is essentially the stable orbifold cohomology of the moduli stack of curves over the complex numbers.

The conjecture was proven in (Madsen-Weiss 02).

Exposition and review is in (Madsen 07).

References

The conjecture is due to

• David Mumford, Towards an enumerative geometry of the moduli space of curves, In Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, MA, 1983, 271–328.

The proof is due to

• 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

See also

• Geoffrey Powell, The Mumford conjecture (after Madsen and Weiss), Séminaire Bourbaki. Vol. 2004/2005. Astérisque 307 (2006), Exp. No. 944, viii, 247–282, MR2009f:55018

• Michael Weiss, Cohomology of the stable mapping class group, Topology, geometry and quantum field theory, 379–404, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press 2004, MR2005e:57059

• Ib Madsen, Ulrike Tillmann, The stable mapping class group and $Q(\mathbf{C} P^\infty_+)$, Invent. Math. 145 (2001), no. 3, 509–544. MR2002h:55011

Review and exposition is in

• Ib Madsen, Moduli spaces from a topological viewpoint, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) (pdf)