In our previous work we have defined the notion of characteristic classes of surface bundles, which are differentiable fibre bundles whose fibres are closed oriented surfaces. In this paper we derive new relations between these characteristic classes by considering a canonical embedding of a given surface bundle with cross section to its associated family of Jacobian manifolds. As a key technical step we determine the first cohomology group of the mapping class group of oriented surfaces with coefficients in the homology of the surface. This might have an independent meaning.
Dans notre travail précédent nous avons défini la notion de classes caractéristiques des fibrés en surface, qui sont des espaces fibrés différentiables dont les fibres sont des surfaces fermées orientées. Dans cet article nous dérivons de nouvelles relations entre ces classes caractéristiques en considérant un plongement canonique d’un fibré en surface donné avec section dans sa famille associée de variétés jacobiennes. Comme étape technique cruciale nous déterminons la première cohomologie du groupe de difféotopies d’une surface orientée à valeurs dans l’homologie de la surface. Ceci peut avoir un sens indépendant.
@article{AIF_1989__39_3_777_0, author = {Morita, Shigeyuki}, title = {Families of jacobian manifolds and characteristic classes of surface bundles. {I}}, journal = {Annales de l'Institut Fourier}, pages = {777--810}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {39}, number = {3}, year = {1989}, doi = {10.5802/aif.1188}, zbl = {0672.57015}, mrnumber = {91d:32028}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1188/} }
TY - JOUR AU - Morita, Shigeyuki TI - Families of jacobian manifolds and characteristic classes of surface bundles. I JO - Annales de l'Institut Fourier PY - 1989 SP - 777 EP - 810 VL - 39 IS - 3 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1188/ DO - 10.5802/aif.1188 LA - en ID - AIF_1989__39_3_777_0 ER -
%0 Journal Article %A Morita, Shigeyuki %T Families of jacobian manifolds and characteristic classes of surface bundles. I %J Annales de l'Institut Fourier %D 1989 %P 777-810 %V 39 %N 3 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1188/ %R 10.5802/aif.1188 %G en %F AIF_1989__39_3_777_0
Morita, Shigeyuki. Families of jacobian manifolds and characteristic classes of surface bundles. I. Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 777-810. doi : 10.5802/aif.1188. https://aif.centre-mersenne.org/articles/10.5802/aif.1188/
[BH] On mapping class groups of closed surfaces as covering spaces, In : Advances in the theory of Riemann surfaces, Ann. Math. Studies, 66(1971), 81-115. | MR | Zbl
and ,[Br] Cohomology of groups, Springer, 1982. | MR | Zbl
,[D] Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology, 15 (1976), 233-245. | MR | Zbl
,[E] Families of Riemann surfaces and Jacobi varieties, Ann. of Math., 107 (1978), 255-286. | MR | Zbl
,[G] Lectures on Riemann surfaces — Jacobi varieties, Math. Notes, 12, Princeton Univ. Press, 1972. | MR | Zbl
,[H] The second homology group of the mapping class of group of an orientable surface, Invent. Math., 72 (1983), 221-239. | MR | Zbl
,[HS] Cohomology of group extensions, Trans. Amer. Math. Soc., 74 (1953), 110-134. | MR | Zbl
and ,[J] A survey of the Torelli group, Contemp. Math., 20 (1983), 165-179. | MR | Zbl
,[L] A finite set of generators for the homeotopy group of a 2-manifold, Proc. Camb. Phil. Soc., 60 (1964), 769-778; ibid. 62 (1966), 679-681. | MR | Zbl
,[Mi] Homology of the mapping class group, J. Diff. Geom., 24 (1986), 1-14. | MR | Zbl
,[Mil] On the existence of a connection with zero curvature, Comment. Math. Helv., 32 (1958), 215-223. | MR | Zbl
,[Mo1] Nontriviality of the Gelfand-Fuchs characteristic classes for flat S1-bundles, Osaka J. Math., 21 (1984), 545-563. | MR | Zbl
,[Mo2] Characteristic classes of surface bundles, Invent. Math., 90 (1987), 551-577. | MR | Zbl
,[Mo3] Characteristic classes of surface bundles and bounded cohomology, in “ A Fête of Topology”, 233-257, Academic Press, 1988. | MR | Zbl
,[Mo4] Characteristic classes of T2-bundles I, II, Advanced Studies in Pure Math., 9 (1986), 135-148 and to appear. | MR | Zbl
,[Mo5] Family of Jacobian manifolds and characteristic classes of surface bundles, Proc. Japan Acad., 60 (1984), 373-376. | MR | Zbl
,[Mo6] Families of Jacobian manifolds and characteristic classes of surface bundles II, to appear in Math. Proc. Camb. Phil. Soc. | Zbl
,[Mo7] Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I, to appear in Topology. | Zbl
,[Mu1] Curves and their Jacobians, The University of Michigan Press, 1975. | MR | Zbl
,[Mu2] Towards an enumerative geometry of the moduli space of curves, In : Arithmetic and Geometry, Progress in Math., 36 (1983), 271-328. | MR | Zbl
,[S] Infinitesimal computations in topology, Publ. Math. I.H.E.S., 47 (1977), 269-331. | Numdam | MR | Zbl
,Cited by Sources: