The family index of the odd signature operator with coefficients in a flat bundle

We study characteristic classes arising as the indices of families of elliptic operators acting on the fibers of an oriented M-bundle f : E -->B, M a smooth oriented closed manifold. Given a family of such operators D = {Db} one obtains a family index Ind(D) in K^*(B). If D is "sufficiently natural" (in a sense made precise in [23]) these indices may be viewed as arising from certain universal symbol classes �in K^*(MTSO(n)), where MTSO(n) is the Thom spectrum of the additive inverse of the universal bundle of oriented n-planes over BSO(n). Explicitly, Ind(D) is pulled back from K^*(MTSO(d)) by the Madsen-Tillman-Weiss map associated to f : E --> B. We show Ind(DVo ) = 0 where DVo is the family of odd signature operators on the fibers of f : E --> B with coefficients in a flat Hermitian vector bundle V --> E. DVo is not universal in the sense of [23] however its index can be described in terms of universal symbols. The vanishing relations implied in cohomology show the higher signatures (Novikov [52]) associated to flat Hermitian bundles provide obstructions to fibering as an odd-dimensional manifold bundle. We end by discussing some examples of flat Hermitian vector bundles to verify that these higher signatures provide a more general obstruction than the usual signature of a 4k-dimensional manifold.