# A computation of K-theory cochains and its applications

### Abstract

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:

• On page 16, $\mathcal{C}\to\mathcal{C} \to \mathcal{C}$ should read $\mathcal{C}\times \mathcal{C} \to \mathcal{C}$.
• On page 18, $N(\mathcal{C})^\otimes \to N(\mathcal{C})$ should read $N(\mathcal{C})^\otimes \to \mathbf{Fin}_*$.
• In Proposition 1.73, $\mathscr{S}$ should read $\mathscr{S}^{\mathrm{op}}$.
• The proof of Proposition 1.81 is incorrect.
• In Theorem 3.17, $H^*(X;\mathbf{Q})$ and $H^*(X;\mathbf{Z}_{(p)})$ should be replaced by $H_*(X;\mathbf{Q})$ and $H_*(X;\mathbf{Z}_{(p)})$, respectively.
• In the introduction to Chapter 4, the rational cdga’s should be augmented.
• In Proposition 4.33, “K-theoric” should read “K-theoretic”.
• In Chapter 5 (specifically, Theorem 5.15), the condition that $L_{K(1)}\Sigma^\infty_+ X$ should be $K(1)$-locally dualisable is superfluous: this is implied by the condition that $KU_p^*(X) \cong \Lambda_{\mathbf{Z}_p}[M]$ (using Theorem 2.13).
• In Remark 5.14, $\mathscr{M}(M/\theta^p_M)$ should read $\mathscr{M}^\vee(M/\theta^p_M)$.
• In the proof of Lemma 5.18, $h^* \circ \varepsilon$ should have target $KU_p^{X_+}$.