Orientation system of the Mobius band

Give the rectangle the local orientation represented by the ordered coordinate directions $({\mathrm{\partial }}_x,{\mathrm{\partial }}_s)$. The gluing sends these to $({\mathrm{\partial }}_x,-{\mathrm{\partial }}_s)$, so one traversal of the core returns the negative local orientation. By [F1], its monodromy is $-1$.

Unwrap the gluing twice: $[0,2]\times [-1,1]\mathrm{/}(0,s)\sim (2,s)$ maps two-to-one onto $B$. This space is an annulus, and its core maps twice around the core of $B$. Hence the pulled-back monodromy is $(-1{)}^2=1$. Since the annulus retracts onto its core circle, the pulled-back rank-one local system is constant.

The two horizontal rectangle edges are joined into one boundary circle. Traversing this circle runs from $(0,1)$ to $(1,1)\sim (0,-1)$ and then from $(0,-1)$ to $(1,-1)\sim (0,1)$, so its image in the core has degree two. Its orientation monodromy is therefore $(-1{)}^2=1$. The collar extension in [F2] gives the same transport at boundary points, hence the restricted system is constant. The boundary circle is orientable even though $B$ is not. Empty-boundary and zero-ring cases are not part of this fixed example, and no AC is used.