Homology, Homotopy and Applications

On the 2-adic $K$-localizations of $H$-spaces

A. K. Bousfield

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We determine the 2-adic $K$-localizations for a large class of $H$-spaces and related spaces. As in the odd primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary $v_1$-periodic homotopy groups of our spaces. The present $v_1$-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic $K$-cohomology algebras (including the 2-adic $KO$-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic $K$-cohomology algebras with specified operations.

Article information

Homology Homotopy Appl., Volume 9, Number 1 (2007), 331-366.

First available in Project Euclid: 5 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 55P60: Localization and completion 55Q51: $v_n$-periodicity 55S25: $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx]

$K$-localizations $v_1$-periodic homotopy 2-adic $K$-theory united $K$-theory compact Lie groups


Bousfield, A. K. On the 2-adic $K$-localizations of $H$-spaces. Homology Homotopy Appl. 9 (2007), no. 1, 331--366. https://projecteuclid.org/euclid.hha/1175791099

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