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

TL;DR

This paper investigates the B"okstedt-Madsen map from the cobordism category to $A$-theory, showing it extends the parametrized $A$-theory Euler characteristic and factors through a canonical map, linking cobordism and algebraic $K$-theory.

Contribution

It establishes that the B"okstedt-Madsen map extends the universal parametrized $A$-theory Euler characteristic and factors through the canonical unit map, connecting cobordism and $A$-theory.

Findings

The map extends the universal parametrized $A$-theory Euler characteristic.

It factors through the canonical unit map $Q(BO(d)_+) o A(BO(d))$.

Links cobordism category to algebraic $K$-theory via $A$-theory.

Abstract

B\"okstedt 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 $Q(BO(d)_+) \to A(BO(d))$.