Lift
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
www.almerja.com
الجزء والصفحة:
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15-5-2021
2297
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 
.
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