Compute the fundamental group \(\pi_1(S^1 \vee S^1, x_0)\) of the wedge sum of two circles (the figure eight space).
Solution
Let \(X=S^1_a\vee S^1_b\) with wedge point \(x_0\).
1. Choose points \(p_a\in S^1_a\setminus\{x_0\}\) and \(p_b\in S^1_b\setminus\{x_0\}\), and put\[ U=X\setminus\{p_b\}, \qquad V=X\setminus\{p_a\}. \]Then \(U\simeq S^1_a\), \(V\simeq S^1_b\), and \(U\cap V\) is contractible.
2. Therefore\[ \pi_1(U,x_0)\cong\mathbb Z, \qquad \pi_1(V,x_0)\cong\mathbb Z, \qquad \pi_1(U\cap V,x_0)=1. \]
3. Seifert–van Kampen gives\[ \pi_1(S^1\vee S^1,x_0) \cong \mathbb Z*\mathbb Z \cong F_2. \]Thus the figure eight has free fundamental group on the two circle loops.