Orientation line bundle ($\det (TM)$)

The orientation line bundle of a smooth $n$-manifold is the real line bundle ${\bigwedge }^{n}TM$. Its transition functions have the signs of the tangent-coordinate transition determinants. A nonzero continuous section specifies an orientation of the manifold. Its orientation double cover records the two possible local orientations.

The total space of the orientation line bundle is orientable even when its base is not: an orientation reversal in a base coordinate transition is matched by a reversal in the line coordinate, so their product has positive sign. Fiberwise multiplication by $1-t$ gives a deformation retraction onto the zero section. For the real projective plane this total space is $(S^2\times \mathbb{R})/((v,t)\sim (-v,-t))$, an oriented noncompact three-manifold homotopy equivalent to the plane.