A Covering Space Lifting Problem Workshop

A Reliable Order for Lifting Problems

A lifting problem asks whether a map \(f:X\to Y\) can be written as \(f=p\circ\widetilde f\), where \(p:E\to Y\) is a covering map and \(\widetilde f:X\to E\) is the proposed lift. The most efficient approach is not to begin with formulas. First check that the specified points are compatible with \(p\); then decide whether the problem is local, as for a path, or global, as for a map from a space with nontrivial loops. Finally, verify the lift by composing with \(p\), and check endpoints when the domain is an interval.

For a based problem, the data include \(x_0\in X\), \(y_0\in Y\), and \(e_0\in E\), with \(f(x_0)=y_0\). A lift based at \(e_0\) must also satisfy \(\widetilde f(x_0)=e_0\). Necessarily \(p(e_0)=y_0\), since \(p(\widetilde f(x_0))=f(x_0)\). This elementary check can rule out a lift before any further calculation.

Proof. Let \(D=\{x\in X:\widetilde f_1(x)=\widetilde f_2(x)\}\). We show that \(D\) and \(X\setminus D\) are open. Fix \(x\in X\), and choose an evenly covered neighborhood \(U\) of \(f(x)\). By continuity, there is an open neighborhood \(V\) of \(x\) with \(f(V)\subseteq U\). Each lift on \(V\) lies in one sheet over \(U\) after restricting to a sufficiently small neighborhood of \(x\): the sheet containing its value is open, and its inverse under the lift is an open neighborhood of \(x\). On that neighborhood, the lift is the inverse of \(p\) on that sheet composed with \(f\). If the two lift values at \(x\) agree, the sheets are the same, so these inverse formulas agree nearby. If the values differ, they lie in different sheets, so the formulas differ nearby. Thus \(D\) and its complement are open. Since \(X\) is connected, \(D\) is either empty or all of \(X\). It is nonempty because the lifts agree at one point, so \(D=X\). \(\square\)

Path Problems: Follow the Sheets

For a path \(\alpha:I\to Y\), where \(I=[0,1]\) has its usual topology, the path-lifting theorem says that for each \(e_0\in p^{-1}(\alpha(0))\), there is a unique lift \(\widetilde\alpha:I\to E\) with \(\widetilde\alpha(0)=e_0\). The theorem applies because \(p\) is a covering map. Evenly covered neighborhoods explain the local calculation: while \(\alpha\) stays in one such neighborhood \(U\), the lift is obtained by applying the inverse of \(p\) on the sheet selected by the current lift value. Subdivision of the interval lets us carry out this procedure along the whole path.

For computations below, let \(C_m=\mathbb{R}/m\mathbb{Z}\) have the quotient topology, and write \([t]_m\) for the class of \(t\). Consider \(p:C_4\to C_2\), \(p([t]_4)=[t]_2\). This map is well defined because changing \(t\) by a multiple of \(4\) does not change its class modulo \(2\). It is a covering map: an arc in \(C_2\) of length less than \(1\) has preimage consisting of two disjoint arcs in \(C_4\), and \(p\) maps each of those arcs homeomorphically onto the original arc.

Map Problems: Apply the Subgroup Test

For a map from a connected, locally path-connected space, checking a few local sheets is not enough: loops in the domain can impose a global obstruction. The fundamental-group criterion handles that obstruction. For a based space \((X,x_0)\), \(\pi_1(X,x_0)\) denotes the group of based loops up to based homotopy, with multiplication by concatenation. A based map \(f:(X,x_0)\to(Y,y_0)\) induces a homomorphism \(f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)\).

This criterion is a standard result established in the earlier material on the fundamental-group criterion for lifting maps. Its hypotheses matter: the base points must match, and the stated connectedness and local path-connectedness assumptions on \(X\) are part of the theorem. The image \(p_*(\pi_1(E,e_0))\) consists of the based loop classes in \(Y\) that arise by projecting based loops in \(E\).

Loop Endpoints as a Direct Test

For a based loop \(\gamma:I\to Y\) with \(\gamma(0)=\gamma(1)=y_0\), lift it from \(e_0\in p^{-1}(y_0)\). The lift need not be a loop. The following criterion relates its endpoint to the subgroup test.

Proof. If the lift \(\widetilde\gamma\) ends at \(e_0\), it is a based loop in \(E\), and \(p\circ\widetilde\gamma=\gamma\). Therefore \([\gamma]=p_*([\widetilde\gamma])\), so \([\gamma]\) belongs to the indicated image. Conversely, suppose \([\gamma]=p_*([\lambda])\) for a based loop \(\lambda\) in \(E\) at \(e_0\). Then \(\gamma\) is based-homotopic to \(p\circ\lambda\). The lift of \(p\circ\lambda\) starting at \(e_0\) is \(\lambda\), by uniqueness of path lifts, and ends at \(e_0\). Endpoint invariance under based path homotopy, established in the earlier material on endpoint invariance, implies that the lift of \(\gamma\) from \(e_0\) has the same endpoint. Hence it ends at \(e_0\). \(\square\)

Common Pitfall: A Formula Does Not Make a Loop

A common error is to write down a path formula and call it a loop without checking its endpoint. For \(f(t)=[3t]_2\) on \(I\), the incorrect calculation is \([3]_2=[0]_2\), which would require \(3\) to be a multiple of \(2\). In fact \(3=2+1\), so \([3]_2=[1]_2\ne[0]_2\). The correction is to compare the classes modulo \(2\) explicitly. If a loop is required, choose a formula whose endpoint difference is a multiple of \(2\), such as \(\gamma(t)=[2t]_2\), and then recompute the lift endpoint from its own formula.

A second useful discipline is to separate the methods. For a path, use path lifting and follow the sheets, then check both projected endpoints. For a map from a connected, locally path-connected domain, first check the base point and then use the subgroup criterion. A continuous bijection or a plausible formula is not, by itself, a lifting argument: the defining identity \(p\circ\widetilde f=f\) and all specified base-point conditions must be verified.