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

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

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Keywords

cobordism category, bivariant $A$–theory, parametrized Euler characteristic

Mathematical Subject Classification 2010

Primary: 19D10, 55R12, 57R90

References

Publication

Received: 14 October 2011

Revised: 22 June 2013

Accepted: 22 June 2013

Preview posted: 5 December 2014

Published: 9 January 2014

Authors

George Raptis
Institut für Mathematik Universität Osnabrück Albrechtstr. 28a D-49076 Osnabrück Germany

Wolfgang Steimle
Institut for Matematiske Fag University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark

Volume 14, issue 1 (2014)

George Raptis and Wolfgang Steimle

Abstract