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.

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 k-Form 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