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

Abstract

Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory ${\mathrm{KU}}_p^{\ast }(X)$ 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 $X$.

v2: 39 pages, strengthened results in the appendix, added references, incorporated referee comments. To appear in Algebraic & Geometric Topology. v1: 34 pages, including a joint appendix with Max Blans. Comments welcome!