What is the definition of a regular (or Galois) covering space?
State, without proof, a criterion in terms of the fundamental group for a covering map \(p : \tilde X \to X\) to be regular.
Let \(\Theta\) be the topological space formed as the union of a circle and its diameter (so this space looks exactly like the letter \(\Theta\)). Give an example of a covering space of \(\Theta\) that is not regular.
Solution
1. A connected covering \(p:\widetilde X\to X\) is regular (Galois) if its deck-transformation group acts transitively on each fiber.
Equivalently, for any two points of the same fiber there is a deck transformation carrying one to the other.
2. If \(X\) is path connected, locally path connected, and semilocally simply connected, the connected cover corresponding to a subgroup \(H\le\pi_1(X,x_0)\) is regular iff \(H\) is normal in \(\pi_1(X,x_0)\).
This is the standard subgroup criterion in covering-space theory.
3. For \(\Theta\), choose free generators \(a,b\) of\[\pi_1(\Theta)\cong F_2\]and take the connected covering corresponding to the subgroup \(H=\langle a\rangle\).
The graph \(\Theta\) is homotopy equivalent to a wedge of two circles, hence has free fundamental group of rank \(2\). The subgroup-covering correspondence produces a connected graph covering for every subgroup.
4. This covering is not regular.
The subgroup \(H=\langle a\rangle\) is not normal in \(F_2\): for example \(bab^{-1}\notin\langle a\rangle\). By 2, the corresponding cover is nonregular.