Let \(X = A \vee B\) be the wedge of two circles.
Let \(E\) be the space pictured in Figure 81.4; let \(p: E \to X\) wrap each arc \(A_1\) and \(A_2\) around \(A\) and map \(B_1\) and \(B_2\) homeomorphically onto \(B\). Show that \(p\) is a regular covering map.
Determine the group of covering transformations of the covering of \(X\) indicated in Figure 81.5. Is this covering regular?
Repeat (b) for the covering pictured in Figure 81.6.
Repeat (b) for the covering pictured in Figure 81.7.
Solution
Goal: Determine the groups of covering transformations (deck transformations) \(\mathcal{C}(E, p, X)\) and regularity status for the four graph coverings of the figure-eight space \(X = A \vee B\).
1. General principles for covering transformations of graphs:Proof:
1.1. A covering transformation \(h \in \mathcal{C}(E, p, X)\) is a homeomorphism \(h: E \to E\) satisfying \(p \circ h = p\).
1.2. For graph coverings, \(h\) is a directed-graph automorphism of the 1-skeleton \(E\) that preserves edge orientations and edge labels (\(a\)-edges and \(b\)-edges).
1.3. A covering map of degree \(d\) is regular (normal) if and only if \(\mathcal{C}(E, p, X)\) acts transitively on the fiber \(p^{-1}(x_0)\), which is equivalent to \(|\mathcal{C}(E, p, X)| = d = [F_2 : p_*(\pi_1(E))]\).
2. Part (a) (Figure 81.4 - 2-fold covering):Proof:
2.1. The total space \(E\) has 2 vertices \(\{v_1, v_2\}\), two \(a\)-arcs \(A_1, A_2\) connecting \(v_1\) and \(v_2\), and two loops \(B_1\) at \(v_1\) and \(B_2\) at \(v_2\).
2.2. The map \(\tau: E \to E\) swapping \(v_1 \leftrightarrow v_2\), \(A_1 \leftrightarrow A_2\), and \(B_1 \leftrightarrow B_2\) preserves orientations and labels.
2.3. Thus \(\mathcal{C}(E, p, X) = \{\operatorname{id}, \tau\} \cong \mathbb{Z}/2\mathbb{Z}\).
2.4. Since the deck transformation group has order \(2\) equal to the degree of the covering, it acts transitively on the 2-point fiber.
2.5. Therefore, \(p\) is a regular covering map.
3. Part (b) (Figure 81.5 - 3-fold cyclic covering):Proof:
3.1. The total space has 3 vertices \(\{v_1, v_2, v_3\}\) arranged in a 3-fold rotational cyclic symmetry where \(a\) and \(b\) edges cyclically permute the vertices (\(v_1 \to v_2 \to v_3 \to v_1\)).
3.2. The 3-fold cyclic rotation \(\rho\) generates the group of label-preserving automorphisms: \(\mathcal{C}(E, p, X) = \{\operatorname{id}, \rho, \rho^2\} \cong \mathbb{Z}/3\mathbb{Z}\).
3.3. Since \(|\mathcal{C}(E, p, X)| = 3\) equals the degree of the covering, the action on the fiber is transitive.
3.4. Hence, the covering is regular.
4. Part (c) (Figure 81.6 - 3-fold non-regular covering):Proof:
4.1. The 3-fold covering in Figure 81.6 has an asymmetric vertex structure: one vertex \(v_1\) has a \(b\)-loop, while \(v_2\) and \(v_3\) are connected by a pair of \(b\)-edges.
4.2. Any label-preserving automorphism must map \(v_1\) to a vertex carrying a \(b\)-loop, which forces \(h(v_1) = v_1\).
4.3. Because a covering transformation of a connected covering space that fixes a point must be the identity map, \(h = \operatorname{id}_E\).
4.4. Thus the group of covering transformations is trivial: \(\mathcal{C}(E, p, X) = \{\operatorname{id}\}\).
4.5. Since \(|\mathcal{C}(E, p, X)| = 1 < 3\), the action is not transitive, and the covering is not regular.
5. Part (d) (Figure 81.7 - 4-fold covering):Proof:
5.1. The 4-fold covering in Figure 81.7 has 4 vertices with a central 180-degree rotational symmetry \(\sigma\) exchanging opposite vertices and edges.
5.2. No 90-degree rotational symmetry preserves the alternating edge labeling.
5.3. Thus the group of covering transformations is \(\mathcal{C}(E, p, X) = \{\operatorname{id}, \sigma\} \cong \mathbb{Z}/2\mathbb{Z}\).
5.4. Since \(|\mathcal{C}(E, p, X)| = 2 < 4\), the deck group has order strictly less than the index \(4\), so it does not act transitively on the fiber.
5.5. Therefore, the covering is not regular. Q.E.D.