# Open subsets of the n-torus containing no nontrivial loops

Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is nullhomotopic?

• By "nontrivial loops" do you mean not nullhomotopic? – Arun Debray Oct 4 '17 at 4:09
• If the pushforward homomorphism $\pi_1(U,x_0)\to \pi_1(T^n,x_0)$ is a constant homomorphism, then there is a lifting of $U$ to the universal cover of $T^n$. Since the universal cover admits a deformation retract to each of its singleton subsets, that lifting is nullhomotopic. – Jason Starr Oct 4 '17 at 7:58
• @ArunDebray Yes, that's what I mean. – user83520 Oct 4 '17 at 8:42
• @JasonStarr Thank you for that comment. Somehow, I missed this reasoning as I was distracted by the fact that U might not be path connected. – user83520 Oct 4 '17 at 9:08
• I was also assuming that $U$ is path connected. However, you can repeat the argument with any connected component of $U$. – Jason Starr Oct 4 '17 at 10:43