A PARAMETERIZED INDEX THEOREM FOR THE ALGEBRAIC K-THEORY EULER CLASS

Abstract. ARiemann-Roch theorem asserts that some algebraically defined wrong– way map in K-theory agrees with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topo-logically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize their theorem, refine it, and prove a converse stating that an ap-propriate Riemann–Roch equation holds for a compact topological manifold bundle if and only if up to fiber homotopy the bundle has a fiberwise smooth structure. We obtain a similar characterization of fibrations which are fibre homotopy equivalent to compact smooth manifold bundles. 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. The classical setting. Suppose that p: E → B is a smooth fiber bundle with compact fiber F; in other words F is a smooth manifold (possibly with boundary) and the structure group of p is the topological group of diffeomorphisms of F. Le