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?

  • $\begingroup$ By "nontrivial loops" do you mean not nullhomotopic? $\endgroup$ – Arun Debray Oct 4 '17 at 4:09
  • $\begingroup$ 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. $\endgroup$ – Jason Starr Oct 4 '17 at 7:58
  • $\begingroup$ @ArunDebray Yes, that's what I mean. $\endgroup$ – user83520 Oct 4 '17 at 8:42
  • $\begingroup$ @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. $\endgroup$ – user83520 Oct 4 '17 at 9:08
  • $\begingroup$ I was also assuming that $U$ is path connected. However, you can repeat the argument with any connected component of $U$. $\endgroup$ – Jason Starr Oct 4 '17 at 10:43

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