What can you say about the fundamental group of \(X \vee Y\) if \(X\) is homeomorphic to \(S^1\) and \(Y\) is homeomorphic to \(S^2\)?
Solution
1. Well-pointed basepoints and fundamental groups of the summands:
1.1. The circle \(S^1\) and the 2-sphere \(S^2\) are CW complexes, so any choice of basepoints \(x_0 \in S^1\) and \(y_0 \in S^2\) yields a well-pointed pair (the basepoints have contractible open neighborhoods in \(S^1\) and \(S^2\), respectively).
1.2. The fundamental groups of the individual spaces are:\[ \pi_1(S^1, x_0) \cong \mathbb{Z}, \qquad \pi_1(S^2, y_0) \cong \{0\}. \]
2. Application of the Seifert–van Kampen Theorem:
2.1. By the Seifert–van Kampen Theorem for wedge sums of well-pointed spaces:\[ \pi_1(S^1 \vee S^2, p) \cong \pi_1(S^1, x_0) * \pi_1(S^2, y_0). \]
2.2. Substituting the fundamental groups of \(S^1\) and \(S^2\):\[ \pi_1(S^1 \vee S^2, p) \cong \mathbb{Z} * \{0\} \cong \mathbb{Z}. \]
3. Conclusion:The fundamental group of \(S^1 \vee S^2\) is isomorphic to \(\mathbb{Z}\). Q.E.D.