Given a map from a space to a space and another map from a space to a space , a lift is a map from to such that . In other words, a lift of is a map such that the diagram (shown below) commutes.
If is the identity from to , a manifold, and if is the bundle projection from the tangent bundle to , the lifts are precisely vector fields. If is a bundle projection from any fiber bundle to , then lifts are precisely sections. If is the identity from to , a manifold, and a projection from the orientation double cover of , then lifts exist iff is an orientable manifold.
If is a map from a circle to , an -manifold, and the bundle projection from the fiber bundle of alternating n-forms on , then lifts always exist iff is orientable. If is a map from a region in the complex plane to the complex plane (complex analytic), and if is the exponential map, lifts of are precisely logarithms of .