Covering spaces and path lifting

A covering map $p:E\to B$ is a continuous surjection such that every point of B has an open neighborhood U whose inverse image is a disjoint union of open sets, each mapped homeomorphically onto U. These sets are the local sheets. The whole inverse image must have this form; identifying one locally invertible branch near a point is not enough. The circle power map $z\mapsto z^m$ for a positive integer m is a finite covering, while the exponential from the real line is an infinite covering.

A small circle arc avoids the ambiguity of an angle and has one inverse arc on each sheet. By contrast, $z\mapsto z^2$ on the complex plane is branched at zero and is not a covering there: no neighborhood of zero splits into homeomorphic inverse sheets. Removing the branch point from domain and base repairs that example. Covering spaces do not require distances or a differentiable structure. Their definition is topological and includes graph coverings, where local sheets replicate the edge incidence near each vertex.

Given a path gamma in B and a chosen point e over its start, there is a unique lifted path starting at e. Subdivide the compact time interval so successive path pieces lie in evenly covered neighborhoods, and use the unique inverse branch containing the already determined endpoint. Uniqueness on overlaps prevents inconsistent choices. A loop in the base need not lift to a loop: its endpoint can be a different point of the same fiber. Homotopies of paths with fixed endpoints lift so that this endpoint remains unchanged when the initial lift is fixed.

Projecting a nullhomotopy of a lifted loop is immediate; lifting a nullhomotopy in the base shows the induced map on fundamental groups is injective. For a path-connected cover of a path-connected base, the number of sheets is the index of the image subgroup in the base group. These conclusions concern an existing cover and should be separated from the stronger local hypotheses needed later to construct covers from arbitrary subgroups. Path lifting itself follows from the evenly covered definition, not from an assumed universal cover.

For a based map $f:(Y,y_0)\to(B,b_0)$ with Y path connected and locally path connected, and an existing cover with chosen $e_0$ over $b_0$, a based lift exists precisely when $f_*\pi_1(Y,y_0)$ is contained in $p_*\pi_1(E,e_0)$. To construct it, choose paths from $y_0$ to each point, lift their images, and take endpoints. The subgroup condition makes this independent of the chosen paths; local path connectedness supplies the continuity step. Once the starting lift is fixed, uniqueness follows from connectedness and local sheet separation.

For maps between circles, this criterion becomes divisibility of winding degree. A lift through the mth-power cover exists exactly when the map's degree is a multiple of m. One must still specify a lift of the basepoint; different choices can give different lifted maps. The criterion does not say that every loop with a fractional-looking angle formula fails to be continuous, since such a formula may be perfectly continuous on the interval yet fail to close when its endpoints are identified into a circle.