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 
.
