Cobordism categories, bivariant A-theory and the A-theory characteristic

3 March 2014
The A-theory characteristic of a fibration is a map to Waldhausen's algebraic K-theory of spaces which can be regarded as a parametrized Euler characteristic of the fibers. Regarding the classifying space of the cobordism category as a moduli space of smooth manifolds, stable under extensions by cobordisms, it is natural to ask whether the A-theory characteristic can be extended to the cobordism category. A candidate such extension was proposed by Bökstedt and Madsen who defined an infinite loop map from the d-dimensional cobordism category to the algebraic K-theory of BO(d). I will discuss the connections between this map, the A-theory characteristic and the smooth Riemann-Roch theorem of Dwyer, Weiss and Williams.