A computation of K-theory cochains and its applications


Let $KU$ denote the $\mathbf{E}_{\infty}$-ring spectrum of complex K-theory. Let $p$ be an arbitrary prime number, and let $KU_p$ denote the $p$-completion of $KU$. Let $X$ be a pointed space whose K-theory $KU_p^*(X)$ is an exterior algebra on a finite number of odd generators. (This includes odd spheres and many H-spaces.) We give a presentation for the $KU_p$-algebra spectrum $KU_p^{X_+}$ as the cofibre of a map between two ($K(1)$-localised) symmetric $KU_p$-algebra spectra. Combined with previous work of Bousfield, we use this result to show that if $p$ is odd and $X$ satisfies some additional conditions, then the algebra $\mathbf{S}_{K(1)}^{X_+}$ models the $v_1$-periodic homotopy type of $X$. More precisely, we show that the Behrens–Rezk comparison map from the Bousfield–Kuhn functor $\Phi_1 X$ to the $K(1)$-local TAQ-cohomology of $\mathbf{S}_{K(1)}^{X_+}$ is an equivalence.

This is my Master’s thesis, which was supervised by Gijs Heuts.

There are some errata for this thesis:

Sven van Nigtevecht
PhD student