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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
.
See also Lifting Problem