Orientability

Summary

A manifold is orientable when one can choose an orientation at every point, varying continuously over the whole manifold. An orientation is a sense of "clockwise" in dimension $2$, or of "right-handed" in dimension $3$. The sphere and the torus are orientable; the Möbius strip, the Klein bottle and the real projective plane are not. Orientability is needed to integrate differential forms of top degree, to define the fundamental class and to state Poincaré duality with integer coefficients.