A Parametrized Index Theorem for the Algebraic K-theory Euler Class

A Riemann–Roch theorem is a theorem which asserts that some algebraically defined wrong–way map in K –theory agrees or is compatible with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topologically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize and refine their theorem. In the process, we prove a family index theorem for fiber bundles with compact topological manifold fibers, a theorem in which the relevant index equation involves algebraic K –theory. Our methods enable us to make a “universal” choice of algebraic K –theory for such bundles. With this choice, we obtain index–theoretic characterizations of bundles of compact topological manifolds and bundles of compact smooth manifolds, respectively, among fibrations with finitely dominated fibers.