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

Abstract

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.

Algebraic theories of power operations

“…Work of Bousfield [17, 18] describes the $$v_1$$‐periodic homotopy groups of nice spaces in terms of their $$p$$‐adic $$K$$‐theory. One obstruction to extending this to higher heights using $$\operatorname{TAQ}$$ is in determining when $$\Phi _h X \simeq \operatorname{TAQ}_{\mathbb {S}_{K(h)}}(\mathbb {S}_{K(h)}^{X_+})$$; we will not consider this issue here.…”

“…Work of Bousfield [17,18] describes the 𝑣 1 -periodic homotopy groups of nice spaces in terms of their 𝑝-adic 𝐾-theory. One obstruction to extending this to higher heights using TAQ is in determining when Φ ℎ 𝑋 ≃ TAQ 𝕊 𝐾(ℎ) (𝕊 𝑋 + 𝐾(ℎ) ); we will not consider this issue here.…”

The Real K-Theory of Compact Lie Groups

“…He was unable to obtain a complete description of the ring structure, however, and could only make some conjectures about it. In [7], Bousfield determined functorially the united 2-adic Kcohomology algebra of any compact, connected and simply-connected Lie group, which includes the 2-adic KO-cohomology algebra, and hence extended Seymour's results in the 2-adic case, if the Lie group involution is taken into account appropriately.…”

“…The author would like to thank Professor Reyer Sjamaar for suggesting this problem, painstakingly proofreading the manuscript, his patient guidance and encouragement throughout the course of this project. He also thanks the referees for their critical comments, pointing out the relevance of the work [7] and a mistake in the definition of ϕ(dρ) in [9] to him.…”