Can any co-H space be represented as a suspension?

Question

Given a co-H space $Y$, does there exist a space $X$ such that $Y\simeq \Sigma X$? Dually, given a H space $Z$, does there exist a space $W$ such that $Z\simeq \Omega W$? If not, any other relations are also okay.

Answer 1

There are co-H spaces that are not suspensions; for example take an element of order 3 in $\pi_6(S^3)$and form a 2-cell complex. Then this is a co-H space that is not a suspension. For more details, see the article on co-H structures by Arkowitz in the Handbook of Algebraic Topology.