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B\"okstedt and Madsen defined an infinite loop map from the embedded$d$-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss tothe algebraic $K$-theory of $BO(d)$ in the sense of Waldhausen. The purpose ofthis paper is to establish two results in relation to this map. The firstresult is that it extends the universal parametrized $A$-theory Eulercharacteristic of smooth bundles with compact $d$-dimensional fibers, asdefined by Dwyer, Weiss and Williams. The second result is that it actuallyfactors through the canonical unit map $Q(BO(d)_+) \to A(BO(d))$.

On the map of B\"okstedt-Madsen from the cobordism category to $A$-theory