TITLE: Is it known whether this space is a suspension space?
QUESTION [7 upvotes]: For each prime $p\geq 3$ let $\alpha_p:S^{2p}\to S^3$ denote a representative of $\pi_{2p}S^3$ of order $p$. Berstein and Hilton showed that for each $p$ the homotopy cofiber $C_{\alpha_p}$ of $\alpha_p$ is a co-H-space which does not have the homotopy type of a suspension space. 
The maps $\alpha_p$ give rise to a map $\alpha:\bigvee_{p\geq 3} S^{2p}\to S^3$ whose cofiber $C_\alpha$ is a co-H-space. Is it known whether $C_\alpha$ has the homotopy type of a suspension space?

REPLY [10 votes]: Isn't the localization of a 2-connected suspension also a suspension?  Then this cannot be a suspension.  (Brayton Gray pointed this out to me.)