The Mumford conjecture

Powell, Geoffrey

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 945, pp. 247-282.

La conjecture de Mumford affirme que la cohomologie à coefficients rationnels de l’espace de modules stable des surfaces de Riemann est une algèbre de polynômes sur les classes de Mumford-Morita-Miller ; on peut la reformuler en termes de la cohomologie de l’espace classifiant $B{\Gamma }_{\infty }$ construit à partir des groupes modulaires de Teichmüller. La conjecture admet une généralisation topologique, inspirée du théorème de Tillmann que $B{\Gamma }_{\infty }$ devient un espace de lacets infinis après application de la construction plus de Quillen. Le texte présente la démonstration par Madsen et Weiss de la conjecture de Mumford généralisée.

The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space $B{\Gamma }_{\infty }$ derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that $B{\Gamma }_{\infty }$ admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and Weiss of the generalized Mumford conjecture.

Classification : 32G15, 57R20, 55R40, 55R65, 55P15
Keywords: conjecture de Mumford, espace de modules des courbes, groupe modulaire de Teichmüller, théorie de Morse, stratification
Mot clés : Mumford conjecture, moduli space of curves, mapping class group, Morse theory, stratification

[1] T. Bröcker & K. Jänich - Introduction to differential topology, Cambridge University Press, Cambridge, 1982. | MR

[2] C. J. Earle & J. Eells - “A fibre bundle description of Teichmüller theory”, J. Differential Geom. 3 (1969), p. 19-43. | DOI | MR | Zbl

[3] C. J. Earle & A. Schatz - “Teichmüller theory for surfaces with boundary”, J. Differential Geom. 4 (1970), p. 169-185. | DOI | MR | Zbl

[4] S. Galatius - “Mod $p$ homology of the stable mapping class group”, Topology 43 (2004), no. 5, p. 1105-1132. | MR | Zbl

[5] R. Hain & E. Looijenga - “Mapping class groups and moduli spaces of curves”, in Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., vol. 62, Providence, RI, 1997, p. 97-142. | MR | Zbl

[6] J. Harer - “The second homology group of the mapping class group of an orientable surface”, Invent. Math. 72 (1983), no. 2, p. 221-239. | EuDML | MR | Zbl

[7] -, “Stability of the homology of the mapping class groups of orientable surfaces”, Ann. of Math. (2) 121 (1985), no. 2, p. 215-249. | MR | Zbl

[8] J. Harris & I. Morrison - Moduli of curves, Graduate Texts in Math., vol. 187, Springer-Verlag, New York, 1998. | MR | Zbl

[9] N. V. Ivanov - “Stabilization of the homology of Teichmüller modular groups”, Algebra i Analiz 1 (1989), no. 3, p. 110-126. | MR | Zbl

[10] I. Madsen & R. J. Milgram - The classifying spaces for surgery and cobordism of manifolds, Annals of Math. Studies, vol. 92, Princeton University Press, Princeton, NJ, 1979. | MR | Zbl

[11] I. Madsen & U. Tillmann - “The stable mapping class group and $Q\left(ℂ{\mathrm{P}}_{+}^{\infty }\right)$”, Invent. Math. 145 (2001), no. 3, p. 509-544. | MR | Zbl

[12] I. Madsen & M. Weiss - “The stable moduli space of Riemann surfaces: Mumford's conjecture”, Preprint, 2004. | MR | Zbl

[13] D. Mcduff & G. Segal - “Homology fibrations and the “group-completion” theorem”, Invent. Math. 31 (1975/76), no. 3, p. 279-284. | DOI | EuDML | MR | Zbl

[14] E. Y. Miller - “The homology of the mapping class group”, J. Differential Geom. 24 (1986), no. 1, p. 1-14. | MR | Zbl

[15] J. Milnor - Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. | DOI | MR | Zbl

[16] S. Morita - “Characteristic classes of surface bundles”, Invent. Math. 90 (1987), no. 3, p. 551-577. | EuDML | MR | Zbl

[17] -, “Structure of the mapping class groups of surfaces: a survey and a prospect”, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, p. 349-406. | MR | Zbl

[18] -, Geometry of characteristic classes, Translations of Mathematical Monographs, vol. 199, American Mathematical Society, Providence, RI, 2001, Translated from the 1999 Japanese original, Iwanami Series in Modern Mathematics. | MR | Zbl

[19] D. Mumford - “Towards an enumerative geometry of the moduli space of curves”, in Arithmetic and geometry, Vol. II, Progress in Math., vol. 36, Birkhäuser, Boston, MA, 1983, p. 271-328. | MR | Zbl

[20] A. Phillips - “Submersions of open manifolds”, Topology 6 (1967), p. 171-206. | DOI | MR | Zbl

[21] D. Quillen - “Elementary proofs of some results of cobordism theory using Steenrod operations”, Adv. in Math. 7 (1971), p. 29-56 (1971). | MR | Zbl

[22] G. Segal - “Categories and cohomology theories”, Topology 13 (1974), p. 293-312. | DOI | MR | Zbl

[23] R. E. Stong - Notes on cobordism theory, Math. Notes, Princeton University Press, Princeton, NJ, 1968. | MR | Zbl

[24] U. Tillmann - “On the homotopy of the stable mapping class group”, Invent. Math. 130 (1997), no. 2, p. 257-275. | MR | Zbl

[25] V. A. Vassiliev - “Topology of spaces of functions without complicated singularities”, Funktsional. Anal. i Prilozhen. 93 (1989), p. 24-36. | Zbl

[26] -, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by B. Goldfarb. | MR | Zbl