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

Abstract

Let Σ g 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: E → X with a cross-section s : X → E a flat T 2 g -bundle π: J → X and a fibre-preserving embedding j : E → J such that the restriction of j to any fibre E p = π −1 ( p )( p ∈ X ) is equivalent to the Jacobi mapping of E p with respect to some conformal structure on it and relative to the base-point s ( p ) ∈ E p . There is a canonical oriented S 1 -bundle over J and the main result of [11] is the identification of the Euler class of the pullback of this S 1 -bundle by the map j as an element of H 2 ( E , ). In this paper we deal with the case of Σ g -bundles without cross-sections. First of all we associate a flat T 2 g -bundle π′: J ′ → X to any oriented Σ g -bundle π: E → X and construct a fibre-preserving embedding j ′: E → J ′ such that the restriction of j ′ to any fibre E p 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.