Suppose \(X\) is a space that is the union of the closed subspaces \(X_1, \ldots, X_n\); assume there is a point \(p\) of \(X\) such that \(X_i \cap X_j = \ts{p}\) for \(i \neq j\). Then we call \(X\) the wedge of the spaces \(X_1, \ldots, X_n\), and write \(X = X_1 \vee \cdots \vee X_n\). Show that if for each \(i\), the point \(p\) is a deformation retract of an open set \(W_i\) of \(X_i\), then \(\pi_1(X, p)\) is the external free product of the groups \(\pi_1(X_i, p)\) relative to the monomorphisms induced by inclusion.
Solution
Goal: Prove that the fundamental group of a wedge sum \(X = X_1 \vee \dots \vee X_n\) of well-pointed spaces (where \(p\) is a deformation retract of an open neighborhood \(W_i \subseteq X_i\)) is isomorphic to the free product \(*_{i=1}^n \pi_1(X_i, p)\) via the Seifert-van Kampen Theorem.
1. Base case (\(n = 2\)): Let \(X = X_1 \cup X_2\) with closed subspaces \(X_1, X_2\) intersecting at \(X_1 \cap X_2 = \{p\}\).Then \(\pi_1(X, p) \cong \pi_1(X_1, p) * \pi_1(X_2, p)\).Proof:
1.1. Define the subsets \(U_1 = X_1 \cup W_2\) and \(U_2 = X_2 \cup W_1\).
1.2. Openness of \(U_1\) and \(U_2\) in \(X\): - The complement \(X \setminus U_1 = X_2 \setminus W_2\).- Since \(W_2\) is open in \(X_2\), \(X_2 \setminus W_2\) is closed in \(X_2\).- Because \(X_2\) is closed in \(X\), \(X_2 \setminus W_2\) is closed in \(X\), so \(U_1\) is open in \(X\).- By identical reasoning, \(U_2\) is open in \(X\).
1.3. Union and Intersection: - \(U_1 \cup U_2 = (X_1 \cup W_2) \cup (X_2 \cup W_1) = X_1 \cup X_2 = X\).- \(U_1 \cap U_2 = (X_1 \cap X_2) \cup W_1 \cup W_2 = W_1 \cup W_2\).
1.4. Homotopy types of \(U_1, U_2,\) and \(U_1 \cap U_2\): - Since \(W_2\) deformation retracts to \(\{p\}\), extending the deformation retraction by the identity on \(X_1\) exhibits \(X_1\) as a deformation retract of \(U_1\).Hence the inclusion \(j_1: X_1 \hookrightarrow U_1\) induces an isomorphism \((j_1)_*: \pi_1(X_1, p) \xrightarrow{\cong} \pi_1(U_1, p)\).- Symmetrically, the inclusion \(j_2: X_2 \hookrightarrow U_2\) induces \((j_2)_*: \pi_1(X_2, p) \xrightarrow{\cong} \pi_1(U_2, p)\).- The intersection \(U_1 \cap U_2 = W_1 \cup W_2\) deformation retracts to \(\{p\}\), so \(U_1 \cap U_2\) is contractible and \(\pi_1(U_1 \cap U_2, p) = \{1\}\).
1.5. Application of Seifert-van Kampen: - By the Seifert-van Kampen Theorem, the amalgamated product over the trivial group reduces to the free product: \[\pi_1(X, p) \cong \pi_1(U_1, p) *_{\pi_1(U_1 \cap U_2, p)} \pi_1(U_2, p) \cong \pi_1(X_1, p) * \pi_1(X_2, p).\]
2. Induction step for \(n \ge 3\): Proof:
2.1. Assume the statement holds for wedges of \(k < n\) spaces.
2.2. Decompose \(X = Y \vee X_n\), where \(Y = X_1 \vee \dots \vee X_{n-1}\).
2.3. \(Y\) is a closed subspace of \(X\), and \(W_Y = \bigcup_{i=1}^{n-1} W_i\) is an open neighborhood of \(p\) in \(Y\) that deformation retracts to \(\{p\}\).
2.4. Applying the \(n=2\) result from 1 gives: \[\pi_1(X, p) \cong \pi_1(Y, p) * \pi_1(X_n, p).\]
2.5. By the inductive hypothesis, \(\pi_1(Y, p) \cong \pi_1(X_1, p) * \dots * \pi_1(X_{n-1}, p)\).
2.6. Thus \(\pi_1(X, p) \cong \pi_1(X_1, p) * \dots * \pi_1(X_n, p)\).
3. Conclusion: \(\pi_1(X, p)\) is the free product \(\bigast_{i=1}^n \pi_1(X_i, p)\) relative to the inclusion homomorphisms.Q.E.D.