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 $BO(d)$ 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 $Q(BO(d{)}_{+})\to A(BO(d))$.

36 pages. This new version contains additional material and various other changes

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