Algebraic & Geometric Topology

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

George Raptis and Wolfgang Steimle

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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 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)+)A(BO(d)).

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 299-347.

Received: 14 October 2011
Revised: 22 June 2013
Accepted: 22 June 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D10: Algebraic $K$-theory of spaces 55R12: Transfer 57R90: Other types of cobordism [See also 55N22]

cobordism category bivariant $A$–theory parametrized Euler characteristic


Raptis, George; Steimle, Wolfgang. On the map of Bökstedt–Madsen from the cobordism category to $A$–theory. Algebr. Geom. Topol. 14 (2014), no. 1, 299--347. doi:10.2140/agt.2014.14.299.

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