The question is in the title : are there spaces X such that the adjoint of the identity on the loop space $\Omega X$, i.e. $\Sigma\Omega X \to X$, is a homotopy equivalence ?



Can the standard map $\Sigma \Omega X \to X$ be a homotopy equivalence?
If $X$ is such a space (a CW complex, say), then it must be a suspension $\Sigma Y$ (as it is the suspension of $y :=\Omega X$). The James splitting gives
$$\Sigma \Omega \Sigma Y \simeq \bigvee\_{n=1}^\infty \Sigma Y^{\wedge n},$$
the wedge of suspensions of smash products, so under your assumption $\Sigma \Omega \Sigma Y \overset{\sim}\to \Sigma Y$ it follows that $\Sigma Y^{\wedge n} \simeq \*$ for all $n > 1$. In particular, by the Kunneth theorem $Y$ must have the homology of a point, and in particular be path-connected. But then $X=\Sigma Y$ must be simply-connected, and also have the homology of a point: therefore it is contractible.


