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Dec 22, 2015 at 15:43 vote accept James Schwass
Dec 22, 2015 at 15:42 comment added James Schwass I gave that a shot, but at some point Berstein and Hilton require a particular map to be injective on homotopy groups; I couldn't convince myself that the analogs of that map that arise in the construction of $C_\alpha$ enjoy this property.
Dec 22, 2015 at 11:11 answer added Robert Bruner timeline score: 10
Dec 22, 2015 at 8:23 comment added Mark Grant Have you tried extending the proof of Lemma 3.6 in Berstein and Hilton to show that if $C_\alpha$ is a suspension then each of the $\alpha_p$ must be suspensions?
Dec 22, 2015 at 0:22 history asked James Schwass CC BY-SA 3.0