A lift of a continuous map 
through a continuous
map 
is a continuous
map 
satisfying
 |
The diagram above expresses this function composition identity.
For a fiber bundle 
, lifts of the identity
map of 
are exactly its bundle
sections. In the smooth setting, lifts of the identity
map of a smooth manifold 
through its tangent bundle
projection are vector fields. Lifts of the identity
map through the orientation cover exist
iff 
is an orientable manifold.
For an 
-dimensional smooth manifold 
, smooth lifts of the identity
map through the line bundle 
are differential
n-forms. The zero section always gives
such a lift. A nowhere-vanishing lift is a volume
form and exists iff 
is an orientable manifold
(Zinger 2011, pp. 69-70).
For a holomorphic map 
on a region 
in the complex plane, a holomorphic lift through the exponential
function 
is a logarithm of 
. Such a lift need not exist globally.
When 
is a covering
map, every path in 
has a unique lift after its initial
point in 
is specified (Hatcher 2002, p. 60). More generally, suppose
that 
is pathwise-connected
and locally pathwise-connected, with
and 
satisfying 
. A lift with 
exists iff
 |
where 
and 
are the induced homomorphisms
of fundamental groups. The lift is then unique
(Hatcher 2002, pp. 61-62).
In graph covering theory, if 
is a graph cover and 
, then 
is called a lift of 
.
