Algebraic & Geometric Topology

The spectrum for commutative complex $K$–theory

Simon Philipp Gritschacher

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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

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

Received: 14 September 2017
Revised: 24 November 2017
Accepted: 16 December 2017
First available in Project Euclid: 22 March 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}
Secondary: 55R35: Classifying spaces of groups and $H$-spaces 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55R50: Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}

$K$–theory classifying space


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.

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