Classification of connected covering spaces

Under the usual local hypotheses, based connected coverings of $X$ correspond to subgroups of ${\pi }_1(X)$; changing the point above the base point conjugates the subgroup. Normal subgroups correspond to regular coverings.

A connected covering is regular when its deck transformation group acts transitively on every fibre. Under the classification of coverings, this is equivalent to the corresponding subgroup of the fundamental group being normal.

For a connected covering corresponding to $H\le {\pi }_1(X)$, every fibre has cardinality $[{\pi }_1(X):H]$.

If the base of a covering is path-connected, a path from $x_0$ to $x_1$ gives a bijectionby sending each initial lift point to the endpoint of its unique lifted path. Lifting the reversed path gives the inverse bijection.

For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.

Choose permutations ${\alpha }_1,\dots ,{\alpha }_r\in S_n$. A covering of a wedge of $r$ oriented circles has fibre $\{1,\dots ,n\}$ and an edge labelled $j$ from $i$ to ${\alpha }_j(i)$ above the $j$th circle. It is connected exactly when the generated permutation group is transitive, and its deck group is the centralizer of that group in $S_n$.

For a connected regular covering corresponding to $H\le {\pi }_1(X,x_0)$, the subgroup $H$ is normal andThe quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.

A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.

A universal covering has simply connected total space, so its associated subgroup of the base fundamental group is trivial. The trivial subgroup is normal; hence every universal covering map is a normal covering map.

For a connected degree-$n$ cover of a closed orientable surface ${\Sigma }_g$, normality is forced when $n=1$, when $n=2$, or when $g=1$. For $g=0$, existence itself forces $n=1$. For every $g\ge 2$ and $n\ge 3$, nonnormal connected degree-$n$ covers exist.

For $g\ge 2$ and $n\ge 3$, sendwhere $\sigma =(12\cdots n)$ and $\tau =(12)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma ,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.