#### Volume 18, issue 2 (2018)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Other MSP Journals
The spectrum for commutative complex $K$–theory

### Simon Philipp Gritschacher

Algebraic & Geometric Topology 18 (2018) 1205–1249
##### Abstract

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 $BU\left(1\right)$ and thus obtain a splitting of its representing space ${B}_{com}U$ as a product of all the terms in the Whitehead tower for $BU\phantom{\rule{0.3em}{0ex}}$, ${B}_{com}U\simeq BU×BU〈4〉×BU〈6〉×\cdots \phantom{\rule{0.3em}{0ex}}$. As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for ${B}_{com}U$ we describe the relationship of our results with a previous computation of the rational cohomology algebra of ${B}_{com}U\phantom{\rule{0.3em}{0ex}}$. This gives an essentially complete description of the space ${B}_{com}U$ introduced by A Adem and J Gómez.

##### Keywords
$K$–theory, classifying space
##### Mathematical Subject Classification 2010
Primary: 55N15
Secondary: 55R35, 55R40, 55R50