On the map of Bokstedt-Madsen from the cobordism category to A-theory

Abstract

Bokstedt and Madsen defined an infinite loop map from the embedded d-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic K-theory of BO(d) in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it extends the universal parametrized A-theory Euler characteristic of smooth bundles with compact d dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map Q(BO(d)(+)) -> A(BO(d)).