The spectrum for commutative complex K–theory

Gritschacher, Simon

doi:10.2140/agt.2018.18.1205

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The spectrum for commutative complex $K$–theory

Simon Philipp Gritschacher

Algebraic & Geometric Topology 18 (2018) 1205–1249

DOI: 10.2140/agt.2018.18.1205

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 $B U$ , $B_{com} U ≃ B U \times B U 〈 4 〉 \times B U 〈 6 〉 \times \dots$ . 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.

Keywords

$K$–theory, classifying space

Mathematical Subject Classification 2010

Primary: 55N15

Secondary: 55R35, 55R40, 55R50

References

Publication

Received: 14 September 2017

Revised: 24 November 2017

Accepted: 16 December 2017

Published: 12 March 2018

Authors

Simon Philipp Gritschacher
Department of Mathematical Sciences University of Copenhagen København Denmark

Volume 18, issue 2 (2018)