## Characteristic classes of surface bundles.(English)Zbl 0608.57020

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

### Citations:

Zbl 0579.57005; Zbl 0618.57005
Full Text:

### 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 [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 [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 [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 [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 [12] [Hz] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457-485 (1986) · Zbl 0616.14017 [13] [Harr] Harris, J.: Families of smooth curves. Duke Math. J.51, 409-419 (1984) · Zbl 0548.14009 [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 [17] [Ko] Kodaira, K.: A certain type of irregular algebraic surfaces. J. Anal. Math.19, 207-215 (1967) · Zbl 0172.37901 [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 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.