The reduced bu Whitehead conjecture

Work at a prime $p$. Let $C(k)$ be the cofiber of the map from $\operatorname{Sp}^{p^k}({\bf S})$ to $A_{p^k}$, and define

$${\overline{M}}(k)=\Sigma^{-(k+1)}C(k)/C(k-1).$$

These spectra form the chain complex

$$\dots\longrightarrow {\overline{M}}(2)\longrightarrow {\overline{M}}(1)\longrightarrow {\overline{M}}(0)\longrightarrow bu.$$

Reduced $bu$ Whitehead conjecture. The complex is exact at the prime $p$.

This is the reduced $bu$-analogue of the classical Whitehead resolution. The paper constructs the spectra and presents evidence from the calculus of functors, but exactness remains conjectural.