The K-theory cochains of H-spaces and height-1 chromatic homotopy theory

van Nigtevecht, Sven

doi:10.2140/agt.2026.26.29

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The K-theory cochains of H-spaces and height-$1$ chromatic homotopy theory

Sven van Nigtevecht

Appendix: Sven van Nigtevecht and Max Blans

Algebraic & Geometric Topology 26 (2026) 29–63

DOI: 10.2140/agt.2026.26.29

Abstract

Fix an odd prime $p$ . Let $X$ be a pointed space whose $p$ -completed K-theory ${KU}_{p}^{*} (X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a generators-and-relations description of the $𝔼_{\infty}$ - ${KU}_{p}$ -algebra spectrum ${KU}_{p}^{X_{+}}$ of ${KU}_{p}$ -cochains of $X$ . To facilitate this construction, we describe a $K (1)$ -local analogue of the Tor spectral sequence for $𝔼_{1}$ -ring spectra. Combined with previous work of Bousfield, this description of the cochains of $X$ recovers a result of Kjaer that the $v_{1}$ -periodic homotopy type of $X$ can be modelled by these cochains. This then implies that the Goodwillie tower of the height- $1$ Bousfield–Kuhn functor converges for such $X$ .

Keywords

K-theory, higher algebra, H-spaces, chromatic homotopy theory, unstable homotopy theory, cochain spectra, topological André–Quillen cohomology

Mathematical Subject Classification

Primary: 55P43

Secondary: 55Q51, 55T99

References

Publication

Received: 14 October 2022

Revised: 13 January 2025

Accepted: 7 February 2025

Published: 16 January 2026

Authors

Sven van Nigtevecht
Mathematisches Institut Universität Bonn Bonn Germany

Max Blans
Mathematical Institute University of Oxford Oxford United Kingdom

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Volume 26, issue 1 (2026)