Maybe it is an easy question but I cannot figure out.
If the fundamental group of (you may assume compact) an $n$-dimensional manifold $M$ is trivial, i.e., $$\pi_1(M)=0,$$ then can we conclude that $M$ is orientable?
2026-03-29 06:02:25.1774764145
Trivial Fundamental Group and Orientation
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If $M$ is non-orientable, the orientable double cover of $M$ gives you a connected cover space of $M$ where each point has a two-point fiber. Thus the fundamental group of $M$ cannot be trivial.