#### Volume 14, issue 1 (2014)

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

### George Raptis and Wolfgang Steimle

Algebraic & Geometric Topology 14 (2014) 299–347
##### Abstract

Bökstedt and Madsen defined an infinite loop map from the embedded $d\phantom{\rule{0.3em}{0ex}}$–dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic $K\phantom{\rule{0.3em}{0ex}}$–theory of $BO\left(d\right)$ 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 $A\phantom{\rule{0.3em}{0ex}}$–theory Euler characteristic of smooth bundles with compact $d\phantom{\rule{0.3em}{0ex}}$–dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map $Q\left(BO{\left(d\right)}_{+}\right)\to A\left(BO\left(d\right)\right)$.

##### Keywords
cobordism category, bivariant $A$–theory, parametrized Euler characteristic
##### Mathematical Subject Classification 2010
Primary: 19D10, 55R12, 57R90