Bökstedt and Madsen defined an infinite loop map from the $d$-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic $K$-theory of $BO(d)$ in the sense of Waldhausen. The goal of the talk will be to explain two results in relation to this map. The first result is that it extends the algebraic $K$-theory Euler class for smooth bundles with compact $d$-dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the stable homotopy of $BO(d)$. This is joint work with G. Raptis.