Characteristic classes of surface bundles. (English) Zbl 0608.57020

Morita, Shigeyuki

Invent. Math. 90, 551-577 (1987).

Let \(\Sigma _ g\) be a closed orientable surface of genus g and let \(M_ g=\pi _ 0(Diff_ + \Sigma _ g)\) be its mapping class group, where \(Diff_ + \Sigma _ g\) is the group of all orientation preserving diffeomorphisms of \(\Sigma _ g\). Also let \(M_{g,*}\) be the mapping class group of \(\Sigma _ g\) relative to the base point. If \(g\geq 2\), then the extension \(\pi _ 1(\Sigma _ g)\to M_{g,*}\to M_ g\) ”classifies” all surface bundles of genus g, namely differentiable fibre bundles with fibre \(\Sigma _ g.\)
In this paper certain cohomology classes \(e\in H^ 2(M_{g,*}; {\mathbb{Z}})\), \(e_ i\in H^{2i}(M_ g; {\mathbb{Z}})\) \((i=1,2,...)\) are defined, and it is proved that the homomorphisms \({\mathbb{Q}}[e_ 1,e_ 2,...]\to H^ *(M_ g; {\mathbb{Q}})\) and \({\mathbb{Q}}[e,e_ 1,e_ 2,...]\to H^ *(M_{g,*}; {\mathbb{Q}})\) are injective up to degree g/3. Here the bound on the degree is based on a result of J. L. Harer [Ann. Math., II. Ser. 121, 215-249 (1985; Zbl 0579.57005)], and the same result for the former homomorphism was also proved independently by E. Y. Miller [J. Differ. Geom. 24, 1-14 (1986)]. A generalization of the above results to the case of surface bundles with cross sections is given and also several relations among characteristic classes of surface bundles are proved.
As a main application, a negative solution to the generalized Nielsen realization problem is given in the following form: the natural surjective homomorphism \(Diff_ + \Sigma _ g\to M_ g\) does not have a right inverse for all \(g\geq 18\).

MSC:

57R22	Topology of vector bundles and fiber bundles
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50	Differential topological aspects of diffeomorphisms
57R20	Characteristic classes and numbers in differential topology
57R30	Foliations in differential topology; geometric theory

Keywords:

closed orientable surface; mapping class group; surface bundles; surface bundles with cross sections; characteristic classes of surface bundles; generalized Nielsen realization problem

Citations:

Zbl 0579.57005; Zbl 0618.57005

References:

[1]	[A] Atiyah, M.F.: The signature of fibre-bundles. Global Analysis, Papers in Honor of K. Kodaira, Tokyo University Press, 1969, pp. 73-84
[2]	[AS] Atiyah, M.F., Singer, I.M.: The index of elliptic operators: IV. Ann. Math.92, 119-138 (1970) · Zbl 0212.28603
[3]	[Bi] Birman, J.S.: The algebraic structure of surface mapping class groups. In: Harvey, W.J. (ed.) Discrete Groups and Automorphic Functions. New York: Academic Press, 1977, pp. 163-198
[4]	[BH] Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces, I. Am. J. Math.80, 458-538 (1958) · Zbl 0097.36401 · doi:10.2307/2372795
[5]	[Bo] Bott, R.: On a topological obstruction to integrability. Global Analysis, Proc. Sympos. Pure Math., Am. Math. Soc.16, 127-131 (1970)
[6]	[EE] Earle, C.J., Eells, J.: The diffeomorphism group of a compact Riemann surface. Bull. Am. Math. Soc.73, 557-559 (1967) · Zbl 0196.09402 · doi:10.1090/S0002-9904-1967-11746-4
[7]	[ES] Earle, C.J., Schatz, A.: Teichmüller theory for surfaces with boundary. J. Differ. Geom.4, 169-185 (1970) · Zbl 0194.52802
[8]	[FH] Farrell, T., Hsiang, W.C.: On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Sympos. Pure Math., Am. Math. Soc.32, 325-337 (1978)
[9]	[Har1] Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math.72, 221-239 (1983) · Zbl 0533.57003 · doi:10.1007/BF01389321
[10]	[Har2] Harer, J.: Stability of the homology of the mapping class groups of orientable surfaces. Ann. Math.121, 215-249 (1985) · Zbl 0579.57005 · doi:10.2307/1971172
[11]	[Har3] Harer, J.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math.84, 157-176 (1986) · Zbl 0592.57009 · doi:10.1007/BF01388737
[12]	[Hz] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457-485 (1986) · Zbl 0616.14017 · doi:10.1007/BF01390325
[13]	[Harr] Harris, J.: Families of smooth curves. Duke Math. J.51, 409-419 (1984) · Zbl 0548.14009 · doi:10.1215/S0012-7094-84-05120-2
[14]	[Hav] Harvey, W.J.: Geometric structure of surface mapping class groups. In: Wall, C.T.C. (ed.) Homological Group Theory. London Mathematical Society Lecture Notes 36, Cambridge University Press 1979, pp. 255-269
[15]	[Hi] Hirzebruch, F.: The signature of ramified coverings. Global Analysis, Papers in Honor of K. Kodaira, Tokyo University Press, 1969, pp. 253-265
[16]	[Ke] Kerckhoff, S.P.: The Nielsen realization problem. Ann. Math.117, 235-265 (1983) · Zbl 0528.57008 · doi:10.2307/2007076
[17]	[Ko] Kodaira, K.: A certain type of irregular algebraic surfaces. J. Anal. Math.19, 207-215 (1967) · Zbl 0172.37901 · doi:10.1007/BF02788717
[18]	[Mi] Miller, E.Y.: The homology of the mapping class group. J. Differ. Geom.24, 1-14 (1986) · Zbl 0618.57005
[19]	[MS] Milnor, J., Stasheff, J.: Characteristic Classes. Ann. Math. Stud. 76, Princeton University Press 1974 · Zbl 0298.57008
[20]	[Mo1] Morita, S.: Characteristic classes of surface bundles. Bull. Am. Math. Soc.11, 386-388 (1984) · Zbl 0579.55006 · doi:10.1090/S0273-0979-1984-15321-7
[21]	[Mo2] Morita, S.: Family of Jacobian manifolds and characteristic classes of surface bundles (preprint); see also Proc. Jpn Acad.60, 373-376 (1984) · Zbl 0573.57009
[22]	[Mo3] Morita, S.: Family of Jacobian manifolds and characteristic classes of surface bundles II (preprint); see also Proc. Jpn Acad.61, 112-115 (1985) · Zbl 0619.57012
[23]	[Mu] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. Arithmetic and Geometry. Progr. Math.36, 271-328 (1983)
[24]	[S] Sullivan, D.: Infinitesimal computations in topology. Publ. Math. I.H.E.S.47, 269-331 (1977) · Zbl 0374.57002
[25]	[T] Thurston, W.: Foliations and groups of diffeomorphisms. Bull. Am. Math. Soc.80, 304-307 (1974) · Zbl 0295.57014 · doi:10.1090/S0002-9904-1974-13475-0