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 .

See also

Bundle Section, Covering Map, Graph Cover, Lifting Problem

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References

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Zinger, A. Notes on Smooth Manifolds and Vector Bundles. Lecture notes, Mar. 23, 2011. https://www.math.stonybrook.edu/~azinger/mat531-spr11/lect1to7.pdf.

Referenced on Wolfram|Alpha

Lift

Cite this as:

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

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