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 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 .