So I came across this exercise: Show that if $Y$ is path-connected then the set of homotopy classes $[I,Y]$ of maps of $I=[0,1]$ into Y has a single point.
But I am confused here. If $Y=S^1$ then not all loops are homotopic! I mean, isn't true that $\pi_1(Y)\subset [I,Y]$? Or maybe I don't understand the set $[I,Y]$ well!