George Raptis: The A-theory Euler characteristic as a bivariant transformation on the parametrized cobordism category

Abstract: 

The A-theory Euler characteristic of a fibration with homotopy finite fibers is a parametrized K-theoretic refinement of the classical Euler characteristic. This can be defined using transfer maps in the algebraic K-theory of spaces (A-theory). The Dwyer-Weiss-Williams index theorem for smooth bundles shows an identification of the A-theory characteristic in terms of the Becker-Gottlieb transfer map and leads to a Riemann-Roch type formula in algebraic K-theory. I will discuss an approach to this index theorem which shows a general identification of the A-theory characteristic regarded as a bivariant transformation. This approach uses a convenient formalism for bivariant theories and known results about the homotopy type of the cobordism category. The method applies also to the corresponding index theorem for topological manifold bundles. If time permits, I will also discuss some related questions about A-theory transfer maps and their interesting connections with properties of Becker-Gottlieb transfer maps.