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Lift


Given a map f from a space X to a space Y and another map g from a space Z to a space Y, a lift is a map h from X to Z such that gh=f. In other words, a lift of f is a map h such that the diagram (shown below) commutes.

Lift

If f is the identity from Y to Y, a manifold, and if g is the bundle projection from the tangent bundle to Y, the lifts are precisely vector fields. If g is a bundle projection from any fiber bundle to Y, then lifts are precisely sections. If f is the identity from Y to Y, a manifold, and g a projection from the orientation double cover of Y, then lifts exist iff Y is an orientable manifold.

If f is a map from a circle to Y, an n-manifold, and g the bundle projection from the fiber bundle of alternating n-forms on Y, then lifts always exist iff Y is orientable. If f is a map from a region in the complex plane to the complex plane (complex analytic), and if g is the exponential map, lifts of f are precisely logarithms of f.


See also

Lifting Problem

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Cite this as:

Weisstein, Eric W. "Lift." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Lift.html

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