On the map of Bökstedt–Madsen from the cobordism category to $A$–theory

Abstract

Bökstedt 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 $\mathit{BO}\left(d\right)$ 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 $Q\left(\mathit{BO}{\left(d\right)}_{+}\right)\to A\left(\mathit{BO}\left(d\right)\right)$.