Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 1 / b / Solution

The zero section $C=\{t=0\}\subset M$ is a circle. Its normal bundle is the real line bundle obtained fromso its clutching map reverses sign once around $C$. Its mod-two Euler class of a real line bundle, equivalently its first Stiefel–Whitney class, therefore satisfiesIf $M$ were orientable, the splittingand the orientation of the circle would orient ${\nu }_C$, forcing this mod-two Euler number to vanish. This contradiction proves that the Möbius band is nonorientable.