A Parametrized Index Theorem for the Algebraic

Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K–theory agree with topologically defined ones [BaDo]. Bismut and Lott [BiLo] proved such a Riemann–Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker–Gottlieb transfer. We generalize and refine their theorem, and prove a converse stating that the Riemann–Roch condition is equivalent to the existence of a fiberwise smooth structure. In the process, we prove a family index theorem where the K–theory used is algebraic K–theory, and the fiber bundles have topological (not necessarily smooth) manifolds as fibers. 0. Introduction This work is inspired by a recent paper of Bismut and Lott [BiLo], especially by their “Riemann–Roch theorem for flat complex vector bundles” . — Suppose that p : E → B is a smooth fiber bundle with compact fiber F . That is, F is a smooth manifold, possibly with boundary, and the structure group of p is the diffeomorphism group of F , a topological group. Let V be a flat complex vector bundle on E, and let Vi be the complex vector bundle on B whose fiber over b ∈ B is the homology group with twisted coefficients Hi(Fb;V ) where Fb is the fiber over b. Then the Atiyah–Singer index theorem for families implies the following equation