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

TL;DR

This paper provides a detailed algebraic description of the cochains of certain H-spaces in height 1 chromatic homotopy theory, connecting K-theory, spectral sequences, and homotopy types.

Contribution

It introduces a generators-and-relations framework for the $ ext{KU}_p$-cochains of specific spaces and describes a $ ext{K}(1)$-local Tor spectral sequence for $ ext{E}_1$-ring spectra.

Findings

Reconstruction of the $ ext{KU}_p$-cochains as an $ ext{E}_ty$-algebra spectrum.

Description of a $ ext{K}(1)$-local Tor spectral sequence for $ ext{E}_1$-ring spectra.

Convergence of the Goodwillie tower of the height 1 Bousfield-Kuhn functor for the spaces considered.

Abstract

Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ 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 $\mathbf{E}_\infty$-$\mathrm{KU}_p$-algebra spectrum $\mathrm{KU}_p^{X_+}$ of $\mathrm{KU}_p$-cochains of $X$. To facilitate this construction, we describe a $\mathrm{K}(1)$-local analogue of the Tor spectral sequence for $\mathbf{E}_1$-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 $X$.