Article
Full entry |
PDF
(0.6 MB)
Feedback
PDF
(0.6 MB)
Feedback
Keywords:
surface bundle; sheaf; classifying space; homological stability
surface bundle; sheaf; classifying space; homological stability
Summary:
The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [23] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.
References:
[1] Bröcker, T., Jänich, K.: Introduction to Differential Topology. Engl. edition, Cambridge University Press, New York (1982); German ed. Springer-Verlag, New York (1973). MR 0674117
[2] Drinfeld, V.: On the notion of geometric realization. arXiv:math/0304064. MR 2119142 | Zbl 1073.55010
[3] Earle, C. J., Eells, J.: A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 19–43. MR 0276999 | Zbl 0185.32901
[4] Earle, C. J., Schatz, A.: Teichmüller theory for surfaces with boundary. J. Differential Geom. 4 (1970), 169–185. MR 0277000 | Zbl 0194.52802
[5] Galatius, S.: Stable homology of automorphism groups of free groups. arXiv:math/0610216.
[6] Galatius, S.: Mod $p$ homology of the stable mapping class group. Topology 43 (2004), 1105–1132. DOI 10.1016/j.top.2004.01.011 | MR 2079997 | Zbl 1074.57013
[7] Galatius, S., Madsen, I., Tillmann, U., Weiss, M.: The homotopy type of the cobordism category. arXiv:math/0605249. MR 2506750
[8] Harer, J.: Stability of the homology of the mapping class groups of oriented surfaces. Ann. of Math. (2) 121 (1985), 215–249. MR 0786348
[9] Ivanov, N.: Stabilization of the homology of the Teichmüller modular groups. Algebra i Analiz 1 (1989), 120–126, translation in Leningrad Math. J. 1 (1990), 675–691. MR 1015128
[10] Madsen, I., Tillmann, U.: The stable mapping class group and $Q\mathbb{C}P^\infty _+$. Invent. Math. 145 (2001), 509–544. DOI 10.1007/PL00005807 | MR 1856399
[11] Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2) 165 (2007), 843–941. MR 2335797 | Zbl 1156.14021
[12] McDuff, D., Segal, G.: Homology fibrations and the “group-completion” theorem. Invent. Math. 31 (1976), 279–284. DOI 10.1007/BF01403148 | MR 0402733 | Zbl 0306.55020
[13] Miller, E.: The homology of the mapping class group. J. Differential Geom. 24 (1986), 1–14. MR 0857372 | Zbl 0618.57005
[14] Milnor, J.: Construction of universal bundles. II. Ann. of Math. (2) 63 (1956), 430–436. DOI 10.2307/1970012 | MR 0077932 | Zbl 0071.17401
[15] Moerdijk, I.: Classifying spaces and classifying topoi. Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995. MR 1440857 | Zbl 0838.55001
[16] Morita, S.: Characteristic classes of surface bundles. Bull. Amer. Math. Soc. 11 (1984), 386–388 11 (1984), 386–388. DOI 10.1090/S0273-0979-1984-15321-7 | MR 0752805 | Zbl 0579.55006
[17] Morita, S.: Characteristic classes of surface bundles. Invent. Math. 90 (1987), 551–557 90 (1987), 551–557. DOI 10.1007/BF01389178 | MR 0914849 | Zbl 0641.57004
[18] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. Arithmetic and Geometry, Vol. II, Progr. in Maths. series 36, 271–328, Birkhäuser, Boston, 1983, pp. 271–328. MR 0717614 | Zbl 0554.14008
[19] Powell, J.: Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc. 68 (1978), 347–350 68 (1978), 347–350. DOI 10.1090/S0002-9939-1978-0494115-8 | MR 0494115 | Zbl 0391.57009
[20] Segal, G.: Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. DOI 10.1007/BF02684591 | MR 0232393 | Zbl 0199.26404
[21] Segal, G.: Categories and cohomology theories. Topology 13 (1974), 293–312. DOI 10.1016/0040-9383(74)90022-6 | MR 0353298 | Zbl 0284.55016
[22] Segal, G.: The topology of spaces of rational functions. Acta Math. 143 (1979), 39–72. DOI 10.1007/BF02392088 | MR 0533892 | Zbl 0427.55006
[23] Tillmann, U.: On the homotopy of the stable mapping class group. Invent. Math. 130 (1997), 257–275. DOI 10.1007/s002220050184 | MR 1474157 | Zbl 0891.55019
[24] Weiss, M.: What does the classifying space of a category classify?. Homology, Homotopy Appl. 7 (2005), 185–195. MR 2175298 | Zbl 1093.57012
Search
Partner of
