Let \(p: \mathbb{R} \to S^1\) be the standard covering map; consider the covering map \(p \times p: \mathbb{R} \times \mathbb{R} \to S^1 \times S^1\). Let \(b_0 = (1, 0) \in S^1\); set \(X = (b_0 \times S^1) \cup (S^1 \times b_0)\); let \(E = (p \times p)^{-1}(X)\); and let \(q: E \to X\) be the covering map obtained by restricting \(p \times p\). The fundamental group of \(X\) has free generators \(\alpha\) and \(\beta\), where \(\alpha\) is represented by a loop in \(b_0 \times S^1\) and \(\beta\) by a loop in \(S^1 \times b_0\). Find a system of free generators for the subgroup \(q_*(\pi_1(E, e_0))\), where \(e_0\) is the origin in \(\mathbb{R}^2\).
Solution
Goal: Find a system of free generators for the subgroup \(H = q_*(\pi_1(E, e_0)) \le \pi_1(X, x_0) \cong F(\alpha, \beta)\) corresponding to the grid covering of the figure-eight space.
1. Identification of the total space and covering map:Proof:
1.1. The base space \(X = (b_0 \times S^1) \cup (S^1 \times b_0) \cong S^1 \vee S^1\) has fundamental group \(\pi_1(X, x_0) = F(\alpha, \beta)\), where \(\alpha\) corresponds to traveling along \(b_0 \times S^1\) (vertical coordinate) and \(\beta\) corresponds to \(S^1 \times b_0\) (horizontal coordinate).
1.2. The total space \(E = (p \times p)^{-1}(X) = (\mathbb{R} \times \mathbb{Z}) \cup (\mathbb{Z} \times \mathbb{R}) \subset \mathbb{R}^2\) is the infinite grid graph with vertices \(V = \mathbb{Z}^2\).
1.3. A path in \(E\) starting at \((0, 0)\) corresponding to a word \(w \in F(\alpha, \beta)\) terminates at \((\operatorname{exp}_\beta(w), \operatorname{exp}_\alpha(w)) \in \mathbb{Z}^2\).
1.4. Thus a loop lifts to a closed path in \(E\) if and only if the total exponent of both \(\alpha\) and \(\beta\) is zero.
1.5. Hence \(H = q_*(\pi_1(E, e_0)) = [F(\alpha, \beta), F(\alpha, \beta)]\) is the commutator subgroup of the free group on two generators.
2. Maximal spanning tree construction in the grid graph \(E\):Proof:
2.1. Let \(T \subset E\) be the tree consisting of the entire horizontal axis \(\mathbb{R} \times \{0\}\) together with all vertical lines \(\{m\} \times \mathbb{R}\) for all \(m \in \mathbb{Z}\):\[T = (\mathbb{R} \times \{0\}) \cup (\mathbb{Z} \times \mathbb{R}).\]
2.2. \(T\) contains all vertices \(\mathbb{Z}^2\) and contains no cycles, so \(T\) is a maximal spanning tree for \(E\).
2.3. The edges of \(E\) not in \(T\) are precisely the horizontal unit segments at non-zero integer heights:\[e_{m, n} = [m, m+1] \times \{n\}, \quad \text{for } m \in \mathbb{Z}, \, n \in \mathbb{Z} \setminus \{0\}.\]
3. Construction of the free basis:Proof:
3.1. By the standard maximal-tree theorem for fundamental groups of 1-complexes (Theorem 84.1), the fundamental group \(\pi_1(E, e_0)\) is freely generated by the simple cycles formed by each edge \(e_{m, n} \notin T\) completed by paths in \(T\) to the basepoint \((0, 0)\).
3.2. The unique simple path in \(T\) from \((0, 0)\) to \((m, n)\) is:\[(0, 0) \xrightarrow{\text{along } \mathbb{R} \times \{0\}} (m, 0) \xrightarrow{\text{along } \{m\} \times \mathbb{R}} (m, n),\]which projects under \(q\) to the word \(\beta^m \alpha^n\).
3.3. Traversing the horizontal edge \(e_{m, n}\) from \((m, n)\) to \((m+1, n)\) projects to \(\beta\).
3.4. The return path in \(T\) from \((m+1, n)\) to \((0, 0)\) traverses \(\{m+1\} \times \mathbb{R}\) down to \((m+1, 0)\) and then \(\mathbb{R} \times \{0\}\) back to \((0, 0)\), projecting to \(\alpha^{-n} \beta^{-(m+1)}\).
3.5. Concatenating these segments yields the loop:\[g_{m, n} = \beta^m \alpha^n \beta \alpha^{-n} \beta^{-(m+1)}.\]
4. Conclusion:A system of free generators for \(q_*(\pi_1(E, e_0))\) is:\[\{ \beta^m \alpha^n \beta \alpha^{-n} \beta^{-(m+1)} \mid m \in \mathbb{Z}, \, n \in \mathbb{Z} \setminus \{0\} \}.\]Q.E.D.