Classification of connected coverings

Let X be path connected, locally path connected, and semilocally simply connected, with chosen $x_0$. Pointed isomorphism classes of path-connected covers $(E,e_0)\to(X,x_0)$ correspond bijectively to subgroups $H\le G=\pi_1(X,x_0)$, assigning $H=p_*\pi_1(E,e_0)$. Isomorphisms must commute with projection and preserve the chosen point. The existence direction uses the universal cover and a quotient by the subgroup action. The uniqueness direction applies the based lifting criterion in both directions and then uses unique lifts to show the resulting maps are inverse.

If the point above $x_0$ is not retained, the classification is by conjugacy classes of subgroups rather than individual subgroups. Paths in the total space between two fiber points project to loops that conjugate the corresponding subgroup. An abstract isomorphism between two subgroups is not enough: their positions inside G matter. Covering isomorphism is also more restrictive than a homeomorphism of total spaces, because the projection must be preserved. Two homeomorphic graphs can cover a wedge of circles in inequivalent ways.

The fiber has cardinality $[G:H]$, possibly infinite. A connected cover is regular when its deck group acts transitively on the fiber, which is equivalent to H being normal. Its deck group is then G/H; for a nonnormal H the correct formula is $N_G(H)/H$. A subgroup can have finite index while its normalizer quotient is much smaller than that index. Finite-sheeted connectedness therefore does not imply a large symmetry group or a regular covering. The trivial subgroup yields the universal cover, while H equal to G yields the identity cover.

A based map between the covers associated to H and K, commuting with the maps to X, exists precisely when $H\subseteq K$. The direction is from the cover with the smaller subgroup to the one with the larger subgroup. This agrees with the universal cover mapping to every other connected cover. If H is contained in K, the number of sheets of that intermediate cover is $[K:H]$. When basepoints are omitted the inclusion condition must be adjusted by a conjugation; choosing a convenient representative cannot be suppressed when comparing concrete subgroups.

Lifting loops permutes the fiber. With our convention of traversing the first loop before the second, this can be written as a right action, avoiding an unnoticed reversal of function composition. The stabilizer of the selected fiber point is H, and connected covers correspond to transitive actions. More generally, orbits identify path components of a covering over this regular base. A deck transformation commutes with the monodromy action. In a transitive action it is determined by the image of one fiber point, but not every candidate image is compatible with every stabilizer.

For circle covers, G is the abelian group of integers, so every subgroup is normal and every connected cover is regular. This special case should not be generalized to a wedge of circles, whose free fundamental group has many nonnormal subgroups. For a finite graph cover, labeling the permutations of the fibers around the graph's generators gives a concrete construction and a practical connectedness test. The abstract subgroup and the permutation description encode the same lifting information, while the geometry still requires local sheets to be attached compatibly along the graph edges.