## Algebraic & Geometric Topology

### The spectrum for commutative complex $K$–theory

Simon Philipp Gritschacher

#### 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 $k u$–group ring of $B U ( 1 )$ and thus obtain a splitting of its representing space $B com U$ as a product of all the terms in the Whitehead tower for $B U$, $B com U ≃ B U × B U 〈 4 〉 × B U 〈 6 〉 × ⋯$. 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 $B com U$. This gives an essentially complete description of the space $B com U$ introduced by A Adem and J Gómez.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 1205-1249.

Dates
Revised: 24 November 2017
Accepted: 16 December 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684035

Digital Object Identifier
doi:10.2140/agt.2018.18.1205

Mathematical Reviews number (MathSciNet)
MR3773753

Zentralblatt MATH identifier
06859619

Keywords
$K$–theory classifying space

#### Citation

Gritschacher, Simon Philipp. The spectrum for commutative complex $K$–theory. Algebr. Geom. Topol. 18 (2018), no. 2, 1205--1249. doi:10.2140/agt.2018.18.1205. https://projecteuclid.org/euclid.agt/1521684035

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