The K(1)-local Spanier-Whitehead dual of KO

Question

Let $D_1KO$ be the $K(1)$-local Spanier-Whitehead dual of $KO$, i.e. the spectrum $$ D_1KO = F(KO,L_{K(1)}S^0). $$

I am interested in what this is. In fact I know that $D_1KO = \Sigma^{-1} KO$. One way to see this is to use the fact that the Gross-Hopkins dual $I_1 = \Sigma^2 P$ for a spectrum $P$ such that $P \wedge KO \simeq \Sigma^4 KO$, along with the equivalence $$ I_1KO = D_1KO \wedge I_1. $$

Here is a claimed direct proof, due to Hahn and Mitchell (see Lemma 8.16).

Start with the cofiber sequence $$ \Sigma^{-1} KO \to \Sigma^{-1} KO \xrightarrow{\delta} L_{K(1)}S^0, $$ and define a map $$ \phi:\Sigma^{-1}KO \to F(KO,\Sigma^{-1}KO) \xrightarrow{\delta_*} F(KO,L_{K(1)}S^0), $$ where the first map is the adjoint to the (desuspension) of the multiplication map. The claim is that $\phi$ induces a weak equivalence on homotopy groups.

How does one show this? Namely, how do we compute $\pi_*F(KO,L_{K(1)}S^0)$?

Answer 1

If everything is completed at an odd prime $p$, and $k$ is a topological generator of $\mathbb{Z}_p^\times$, then $L_{K(1)}S$ is the fibre of $\psi^k-1\colon KU\to KU$. This means that $F(KU,L_{K(1)}S)$ is the fibre of the corresponding self-map $(\psi^k-1)_*$ of $F(KU,KU)$, which is not too hard to understand. A slight adjustment of this should work for $KO$ at $p=2$, but I forget what is the best way to organise it.