Non orientable normal bundle gives a non simply connected manifold

Question

Let $X$ be a compact connected manifold and $M\subset X$ be a compact connected hypersurface. If the normal bundle $NM$ is not orientable, then $\pi_1(X)\not= 0$.

Answer 1

The normal bundle of an hypersurface is a $\mathbb{R}$-bundle which is flat and defined by a representation $\pi_1(M)\rightarrow GL(\mathbb{R})$ if $\pi_1(M)=1$, the normal bundle is trivivial henceforth oriented.

Another way to see this is to take a good trivialisation $U_i$ $(U_i\cap U_j$ is connected) of the normal bundle and consider the chech cocycle $sign(g_{ij})$ where $g_{ij}$ are the coordinate change. It defines a flat bundle which is trivial if and only if the normal bundle is oriented.