For (nice?) pointed spaces, the reduced suspension $\Sigma$ is left adjoint to the loop space $\Omega$. This adjunction is given by the unit maps
$\eta_X : X \to \Omega \Sigma X$, $x \mapsto (t \mapsto [x,t])$
and the counit maps
$\varepsilon_X : \Sigma \Omega X \to X , [\omega,t] \mapsto \omega(t).$
Question. For which $X$ is $\eta_X$ a homotopy equivalence? For which $X$ is $\varepsilon_X$ a homotopy equivalence?
If this is useful, let's assume that $X$ is sufficiently nice (for example a CW complex). You may also replace "homotopy equivalence" by "homology equivalence" etc., if this yields to interesting statements. If there is no characterization: What are interesting classes of examples? And is there any source in the literature where this sort of question is studied?