Suppose $X$ is path connected and $f: S^n \to X$. I am interested in determining what $\pi_1(C_f)$ is.
My guess is that the fundamental group is $\pi_1(X)$ since $C_f = (S^n \times I) \sqcup X /{\sim},$ where $(s,1)\sim f(s)$ and $(s,0)\sim(s',0)$, and the fundamental group of $S^n$ is trivial.