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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!

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Homotopies of loops are required to fix the endpoints: in other words, they are maps $H:I\times I\to Y$ such that $H(0,t)$ and $H(1,t)$ are both the basepoint of the loop for all $t$. But when you just consider homotopy classes of maps $I\to Y$ as in this exercise, there is no condition that the homotopies must fix the endpoints. So, loops which are not homotopic as loops may nevertheless be homotopic without fixing the endpoints.

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  • $\begingroup$ Thanks. So $H(s,0)$ is not necessarily equal to $H(s,1)$ in general. That what I understand. $\endgroup$ – Amrat A Sep 17 '18 at 5:34

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