A PARAMETERIZED INDEX THEOREM FOR THE ALGEBRAIC K-THEORY EULER CLASS W.Dwyer, M.Weiss, B.Williams

A Riemann-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 topologically 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 appropriate 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. 0. Introduction 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 . Let V be a complex vector bundle on E with discrete structure group (flat complex vector bundle for short), and Vi the complex vector bundle on B whose fiber over b ∈ B is the local–coefficient homology group Hi(p (b);V ). Denote by K(Ct) the K-theory space of the topological ring C, so that K(Ct) ≃ Z×BU and for a space X the group of homotopy classes [X,K(Ct)] is the topological complex K-theory of X . The vector bundles V , Vi give classes [V ] ∈ [E,K(Ct)], [Vi] ∈ [B,K(Ct)], and the Atiyah–Singer index theorem for families leads to the equation [BecSch]