The spectrum for commutative complex $K$-theory

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 $BU$, $B_{com}U\simeq BU\times BU⟨4⟩\times BU⟨6⟩\times \dots .$ 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. Gómez.

35 pages. This article replaces "The ring of coefficients for commutative complex $K$-theory". The results have been improved and the exposition streamlined, some results have been removed to appear in future work, some new results have been added, a mistake (previously in Lemma 4.7 and Corollary 4.8) has been corrected (the correct statement appears now as Proposition 5.2)

References in corpus (4)

• A classifying space for commutativity in Lie groups
• Infinite loop spaces and nilpotent K-theory
• Completed representation ring spectra of nilpotent groups
• The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

Cited by in corpus (7)

• Homological stability for spaces of commuting elements in Lie groups
• Classifying spaces for commutativity of low-dimensional Lie groups
• On the mod-$\ell$ homology of the classifying space for commutativity
• Higher generation by abelian subgroups in Lie groups
• Commuting matrices and Atiyah's Real K-theory
• On the second homotopy group of spaces of commuting elements in Lie groups
• Commutative d-Torsion K-Theory and Its Applications