Let \(X\) be the wedge of two circles; let \(p: E \to X\) be a covering map. The fundamental group of \(E\) maps isomorphically under \(p_*\) onto a subgroup \(H\) of the fundamental group of \(X\); the latter is free on two generators \(\alpha\) and \(\beta\).
For each of the four covering spaces \(E\) given in E-79XXG, determine the cardinality of a system of free generators for the fundamental group of \(E\).
For each of these covering spaces, find, in terms of \(\alpha\) and \(\beta\), a system of free generators for the subgroup \(H\) of the fundamental group of \(X\).
Solution
Goal: Determine the rank (cardinality of a free generating set) and explicit generators in terms of \(\alpha, \beta \in \pi_1(X) \cong F_2\) for the subgroups corresponding to the four covering graphs of \(X = S^1 \vee S^1\) in Figures 81.4–81.7.
1. General rank formula for \(d\)-fold coverings of \(S^1 \vee S^1\):Proof:
1.1. \(X\) is a 1-dimensional CW complex with 1 vertex and 2 edges, so \(\chi(X) = 1 - 2 = -1\).
1.2. A connected \(d\)-sheeted covering space \(E\) is a graph with \(V = d\) vertices and \(E = 2d\) edges.
1.3. The Euler characteristic of \(E\) is \(\chi(E) = V - E = d - 2d = -d\).
1.4. Since \(E\) is a connected 1-dimensional CW complex, its fundamental group \(\pi_1(E)\) is a free group of rank:\[\operatorname{rank}(\pi_1(E)) = 1 - \chi(E) = 1 - (-d) = 1 + d.\]
1.5. By the Nielsen-Schreier Theorem, the corresponding subgroup \(H = p_*(\pi_1(E, e_0)) \le F_2\) has index \([F_2 : H] = d\) and free rank \(1 + d\).
2. Part (a): Cardinality of free generating systems.- Fig 81.4 (2-fold covering): Degree \(d = 2 \implies \operatorname{rank} = 1 + 2 = 3\).- Fig 81.5 (3-fold covering): Degree \(d = 3 \implies \operatorname{rank} = 1 + 3 = 4\).- Fig 81.6 (3-fold covering): Degree \(d = 3 \implies \operatorname{rank} = 1 + 3 = 4\).- Fig 81.7 (4-fold covering): Degree \(d = 4 \implies \operatorname{rank} = 1 + 4 = 5\).
3. Part (b): Free generators for \(H \le F_2 = \langle \alpha, \beta \rangle\).Proof:
3.1. Figure 81.4 (\(d = 2\), \(\alpha\) swaps sheets \(0 \leftrightarrow 1\), \(\beta\) loops at each sheet):- Choose maximal tree \(T\) containing the edge \(\tilde{\alpha}_1\) from \(0\) to \(1\).- The non-tree edges yield the 3 free generators:\[H = \big\langle \beta, \; \alpha^2, \; \alpha \beta \alpha^{-1} \big\rangle.\]
3.2. Figure 81.5 (\(d = 3\), \(\alpha\) is 3-cycle \((0\,1\,2)\), \(\beta\) fixes \(0\) and swaps \(1 \leftrightarrow 2\)):- Choose maximal tree \(T = \{0 \xrightarrow{\alpha} 1, \; 1 \xrightarrow{\alpha} 2\}\).- Non-tree edges give the 4 free generators:\[H = \big\langle \beta, \; \alpha^3, \; \alpha \beta \alpha^{-2}, \; \alpha^2 \beta \alpha^{-1} \big\rangle.\]
3.3. Figure 81.6 (\(d = 3\), \(\alpha = (0\,1\,2)\) and \(\beta = (0\,2\,1)\)):- Choose maximal tree \(T = \{0 \xrightarrow{\alpha} 1, \; 1 \xrightarrow{\alpha} 2\}\).- Tracing loops for non-tree edges gives the 4 free generators:\[H = \big\langle \alpha^3, \; \beta^3, \; \alpha \beta, \; \alpha^2 \beta \alpha \big\rangle.\]
3.4. Figure 81.7 (\(d = 4\), \(\mathbb{Z}_2 \times \mathbb{Z}_2\) covering, \(\alpha = (0\,1)(2\,3)\) and \(\beta = (0\,2)(1\,3)\)):- Choose maximal tree \(T = \{0 \xrightarrow{\alpha} 1, \; 0 \xrightarrow{\beta} 2, \; 1 \xrightarrow{\beta} 3\}\).- The remaining 5 non-tree edges give the 5 free generators:\[H = \big\langle \alpha^2, \; \beta^2, \; \alpha \beta \alpha^{-1} \beta^{-1}, \; \alpha \beta^2 \alpha^{-1}, \; \beta \alpha^2 \beta^{-1} \big\rangle.\]Q.E.D.