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Families of Jacobian manifolds and characteristic classes of surface bundles. II

  • Shigeyuki Morita (a1)
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Let Σg be a closed orientable surface of genus g, which will be assumed to be greater than one throughout this paper. In our previous paper [11], we have associated to any oriented ∑g-bundle π: EX with a cross-section s: XE a flat T2g-bundle π: JX and a fibre-preserving embedding j: EJ such that the restriction of j to any fibre Ep = π−1(p)(pX) is equivalent to the Jacobi mapping of Ep with respect to some conformal structure on it and relative to the base-point s(p) ∈ Ep. There is a canonical oriented S1-bundle over J and the main result of [11] is the identification of the Euler class of the pullback of this S1-bundle by the map j as an element of H2(E, ). In this paper we deal with the case of Σg-bundles without cross-sections. First of all we associate a flat T2g-bundle π′: J′ → X to any oriented Σg-bundle π: EX and construct a fibre-preserving embedding j′: EJ′ such that the restriction of j′ to any fibre Ep is equivalent to some translation of the Jacobi mapping of it. Although our original motivation for the present work came from a different source, this should be considered as a topological version of Earle's embedding theorem [5] which states that any holomorphic family of compact Riemann surfaces over a complex manifold can be embedded in a certain associated family of the corresponding Jacobian varieties in an essentially unique way.

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[10]Morita, S.. Characteristic classes of surface bundles. Invent. Math. 90 (1987), 551577.
[11]Morita, S.. Families of Jacobian manifolds and characteristic classes of surface bundles I. (Preprint.)
[12]Morita, S.. Family of Jacobian manifolds and characteristic classes of surface bundles II. Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 112115.
[13]Morita, S.. The second homology group of the mapping class groups of orientable surfaces with twisted coefficients. (In preparation.)
[14]Morita, S.. Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I. (Preprint.)
[15]Mumford, D.. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and Geometry, Progress in Math. no. 36 (Birkhäuser-Verlag, 1983), pp. 271328.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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