I'm learning about loop spaces and the work of Stasheff on $A_{\infty}$-spaces. The broad idea that I'm getting is the following. Given a space $Y$, we want to know under which conditions there exists some space $X$ such that $Y$ is the space of loops of $X$, $Y = \Omega X$ (without previously assuming the existence of such $X$). And such space $X$ exists if and only if $Y$ is an $A_{\infty}$-space (plus some obvious requirements on the homotopy groups of $Y$).
The question that naturally comes to my mind is the following: is there any similar detection tool for reduced suspensions? That is, given a space $X$, is there any general criterion to known whether or not there exists a space $Y$ such that $X$ is the reduced suspension of $Y$, $X = \Sigma Y$? I am assuming all spaces to be pointed.
Many thanks in advance!
