Van Kampen Theorem

Theorem. Let $A, B \subset X$, with intersection $A \overset{\iota_A}{\hookleftarrow} A \cap B \overset{\iota_B}{\hookrightarrow} B$ be open and path-connected with finitely-presented fundamental groups: $$\pi_1(A) \cong \langle a_1, \cdots, a_i\; \vert \; \alpha_1, \cdots, a_n \rangle$$ $$\pi_1(B) \cong \langle b_1, \cdots, b_j\; \vert \; \beta_1, \cdots, \beta_m\rangle$$$$\pi_1(A \cap B) \cong \langle g_1, \cdots, g_k\; \vert \; \gamma_1, \cdots, \gamma_l\rangle$$ Then union $A \cup B$ has fundamental group with presentation: $$\pi_1(A \cup B) = \langle a_1, \cdots, a_i, b_1, \cdots, b_j\; \vert \; \alpha_1, \cdots, \alpha_n, \beta_1, \cdots, \beta_m, \iota_A(g_1)\iota_B(g_1)^{-1}, \cdots, \iota_A(g_k)\iota_B(g_k)^{-1} \rangle$$

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