The spectrum for commutative complex $K$-theory

TL;DR

This paper characterizes the spectrum of commutative complex K-theory, showing it is equivalent to a ku-group ring of BU(1), and provides a detailed splitting and cohomology description of the associated space.

Contribution

It establishes a stable equivalence of the spectrum for commutative complex K-theory to a ku-group ring of BU(1), and describes the space B_{com}U as a product of Whitehead tower terms.

Findings

Spectrum for commutative complex K-theory is equivalent to a ku-group ring of BU(1).

The space B_{com}U splits as a product of Whitehead tower terms.

Provides the ring of coefficients and relates to rational cohomology computations.

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 $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\simeq BU\times BU\langle 4\rangle \times BU\langle 6\rangle \times \dots .$ 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. G\'omez.