Let $X$ be a topological space. And let $\Lambda X=\mathrm{Top}\left[S^1,X\right]$ be the space of continuous loops in $X$. Then how do we calculate $\Pi_0\Lambda X$, the set of connected components of $\Lambda X$. Is it realted to $\pi_1\left(X,x_0\right)$ for some $x_0$?
For simplifications, assume $\Sigma$ is a path connected manifold but I would prefer a general answer.
Suppose X is connected. Since the inclusion of a point into a circle is a cofibration, then the inclusion of pointed loops into free loops is surjective on homotopy classes.
Moreover, the fundamental group of X acts on based loops by conjugation. Hatcher in his appendix proves the orbits of this action are in bijection with unpointed homotopy classes.