## Lift

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 k-Form 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 .