Assume the hypotheses of Theorem 59.1.
What can you say about the fundamental group of \(X\) if \(j_*\) is the trivial homomorphism? If both \(i_*\) and \(j_*\) are trivial?
Give an example where \(i_*\) and \(j_*\) are trivial but neither \(U\) nor \(V\) have trivial fundamental groups.
Solution
Goal: Compute the fundamental group \(\pi_1(X, x_0)\) under Seifert-van Kampen when one or both inclusion homomorphisms \(i_*, j_*\) from \(\pi_1(U \cap V)\) are trivial, and provide an explicit non-trivial example.
1. General Seifert-van Kampen relation structure:Under the hypotheses of Theorem 59.1 (\(X = U \cup V\) with \(U, V, U \cap V\) open and path-connected), the fundamental group is the amalgamated free product:\[\pi_1(X, x_0) \cong (\pi_1(U, x_0) * \pi_1(V, x_0)) / N,\]where \(N\) is the normal closure in \(\pi_1(U) * \pi_1(V)\) of the subset:\[R = \{ i_*(g) j_*(g)^{-1} \mid g \in \pi_1(U \cap V, x_0) \}.\]
2. Analysis of Part (a):Proof:
2.1. Case 1 (\(j_*\) is trivial):- If \(j_*(g) = 1\) for all \(g \in \pi_1(U \cap V, x_0)\), then \(i_*(g) j_*(g)^{-1} = i_*(g) \in \pi_1(U, x_0)\).- The relators lie entirely within the factor \(\pi_1(U, x_0)\), so \(N\) is the normal closure in \(\pi_1(U, x_0)\) of \(i_*(\pi_1(U \cap V, x_0))\).- Factoring out \(N\) affects only the first factor, yielding:\[\pi_1(X, x_0) \cong (\pi_1(U, x_0) / \langle\langle i_*(\pi_1(U \cap V, x_0)) \rangle\rangle) * \pi_1(V, x_0).\]
2.2. Case 2 (Both \(i_*\) and \(j_*\) are trivial):- If \(i_*(g) = 1\) and \(j_*(g) = 1\) for all \(g \in \pi_1(U \cap V, x_0)\), then \(i_*(g) j_*(g)^{-1} = 1\).- Thus the set of relations is \(R = \{1\}\), so the normal subgroup \(N\) is trivial (\(N = \{1\}\)).- Therefore, the amalgamated product reduces directly to the free product:\[\pi_1(X, x_0) \cong \pi_1(U, x_0) * \pi_1(V, x_0).\]
3. Construction for Part (b):Proof:
3.1. Let \(X = S^1 \vee S^1 = A \vee B\) be the wedge sum of two circles with common basepoint \(x_0\).
3.2. Choose points \(p \in A \setminus \{x_0\}\) and \(q \in B \setminus \{x_0\}\).
3.3. Define open subsets:\[U = X \setminus \{q\} = A \vee (B \setminus \{q\}) \quad \text{and} \quad V = X \setminus \{p\} = (A \setminus \{p\}) \vee B.\]
3.4. The open arc \(B \setminus \{q\}\) deformation retracts to \(x_0\), so \(U \simeq A \cong S^1\) and \(\pi_1(U, x_0) \cong \mathbb{Z} \neq \{1\}\).
3.5. Symmetrically, \(V \simeq B \cong S^1\) and \(\pi_1(V, x_0) \cong \mathbb{Z} \neq \{1\}\).
3.6. The intersection \(U \cap V = (A \setminus \{p\}) \vee (B \setminus \{q\})\) is the wedge of two contractible open arcs, hence contractible:\[\pi_1(U \cap V, x_0) = \{1\}.\]
3.7. Since \(\pi_1(U \cap V, x_0)\) is the trivial group, both homomorphisms \(i_*\) and \(j_*\) are trivial, while \(\pi_1(U, x_0) \cong \mathbb{Z}\) and \(\pi_1(V, x_0) \cong \mathbb{Z}\) are non-trivial. Q.E.D.