Suppose that \(i_2\) is surjective.
- Show that \(j_1\) induces an epimorphism
\[ h: \pi_1(U, x_0)/M \to \pi_1(X, x_0), \]
where \(M\) is the least normal subgroup of \(\pi_1(U, x_0)\) containing \(i_1(\ker i_2)\). [Hint: Show \(j_1\) is surjective.]
- Show that \(h\) is an isomorphism. [Hint: Let \(H = \pi_1(U, x_0)/M\). Let \(\phi_1: \pi_1(U, x_0) \to H\) be the projection. Use the fact that \(\pi_1(U \cap V, x_0)/\ker i_2\) is isomorphic to \(\pi_1(V, x_0)\) to define a homomorphism \(\phi_2: \pi_1(V, x_0) \to H\). Use Theorem 70.1 to define a left inverse for \(h\).]
Solution
Goal: Prove that when the inclusion homomorphism \(i_2: \pi_1(U \cap V, x_0) \to \pi_1(V, x_0)\) is surjective, \(\pi_1(X, x_0) \cong \pi_1(U, x_0)/M\), where \(M\) is the normal closure of \(i_1(\ker i_2)\).
1. Part (a): Surjectivity of \(j_1\) and existence of the epimorphism \(h\).Proof:
1.1. By the Seifert-van Kampen theorem, the images of \((j_1)_*\) and \((j_2)_*\) generate \(\pi_1(X, x_0)\), and \((j_1)_* \circ (i_1)_* = (j_2)_* \circ (i_2)_*\).
1.2. For any \(v \in \pi_1(V, x_0)\), surjectivity of \((i_2)_*\) guarantees the existence of \(w \in \pi_1(U \cap V, x_0)\) such that \((i_2)_*(w) = v\).
1.3. Then \((j_2)_*(v) = (j_2)_*((i_2)_*(w)) = (j_1)_*((i_1)_*(w)) \in \operatorname{im}((j_1)_*)\).
1.4. Thus \(\operatorname{im}((j_2)_*) \subseteq \operatorname{im}((j_1)_*)\), which implies \(\operatorname{im}((j_1)_*) = \pi_1(X, x_0)\), so \((j_1)_*\) is surjective.
1.5. For any \(w \in \ker((i_2)_*)\), \((j_1)_*((i_1)_*(w)) = (j_2)_*((i_2)_*(w)) = (j_2)_*(1) = 1\).
1.6. Thus \((i_1)_*(\ker(i_2)_*) \subseteq \ker((j_1)_*)\). Since \(\ker((j_1)_*)\) is normal, the normal closure \(M \subseteq \ker((j_1)_*)\).
1.7. By the First Isomorphism Theorem, \((j_1)_*\) factors through the quotient to define an epimorphism:\[h: \pi_1(U, x_0)/M \to \pi_1(X, x_0), \quad h([u]) = (j_1)_*(u).\]
2. Part (b): \(h\) is an isomorphism (construction of a left inverse via pushout).Proof:
2.1. Let \(H = \pi_1(U, x_0)/M\), and let \(\phi_1: \pi_1(U, x_0) \to H\) be the canonical projection homomorphism.
2.2. Consider the homomorphism \(\phi_1 \circ (i_1)_*: \pi_1(U \cap V, x_0) \to H\). For any \(w \in \ker((i_2)_*)\), \((i_1)_*(w) \in M = \ker(\phi_1)\), so \(\ker((i_2)_*) \subseteq \ker(\phi_1 \circ (i_1)_*)\).
2.3. Since \((i_2)_*: \pi_1(U \cap V, x_0) \to \pi_1(V, x_0)\) is surjective, \(\pi_1(U \cap V, x_0)/\ker((i_2)_*) \cong \pi_1(V, x_0)\).
2.4. Therefore \(\phi_1 \circ (i_1)_*\) factors through \((i_2)_*\) to yield a unique homomorphism \(\phi_2: \pi_1(V, x_0) \to H\) satisfying:\[\phi_2 \circ (i_2)_* = \phi_1 \circ (i_1)_*.\]
2.5. By the universal property of the Seifert-van Kampen theorem (Theorem 70.1), there exists a unique homomorphism \(g: \pi_1(X, x_0) \to H\) such that \(g \circ (j_1)_* = \phi_1\) and \(g \circ (j_2)_* = \phi_2\).
2.6. For any \([u] \in H\):\[(g \circ h)([u]) = g((j_1)_*(u)) = \phi_1(u) = [u].\]
2.7. Thus \(g \circ h = \operatorname{id}_H\), so \(h\) has a left inverse \(g\) and is therefore injective.
2.8. Being both surjective and injective, \(h\) is an isomorphism of groups. Q.E.D.