Volume 18, issue 2 (2018)

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The spectrum for commutative complex $K$–theory

Simon Philipp Gritschacher

Algebraic & Geometric Topology 18 (2018) 1205–1249

We study commutative complex K–theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex K–theory is stably equivalent to the ku–group ring of BU(1) and thus obtain a splitting of its representing space BcomU as a product of all the terms in the Whitehead tower for BU, BcomU BU × BU4× BU6×. As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for BcomU we describe the relationship of our results with a previous computation of the rational cohomology algebra of BcomU. This gives an essentially complete description of the space BcomU introduced by A Adem and J Gómez.

$K$–theory, classifying space
Mathematical Subject Classification 2010
Primary: 55N15
Secondary: 55R35, 55R40, 55R50
Received: 14 September 2017
Revised: 24 November 2017
Accepted: 16 December 2017
Published: 12 March 2018
Simon Philipp Gritschacher
Department of Mathematical Sciences
University of Copenhagen