### A Parametrized Index Theorem for the Algebraic K-Theory Euler Class, by William Dwyer, Michael Weiss, and Bruce Williams

Riemann-Roch theorems assert that certain algebraically defined wrong way
maps (transfers) in algebraic K--theory agree with topologically defined ones
(see Baum, Fulton, MacPherson, Acta Math. 1979). Bismut and Lott
(J. Amer. Math. Soc. 1995) 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.

William Dwyer <Dwyer.1@nd.edu>

Michael Weiss <Weiss.13@nd.edu>

Bruce Williams <Williams.4@nd.edu>