Theorem: Let $X = U \cup V$ with $U, V, U \cap V$ path-connected and $x_0 \in U \cap V$. Then $\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0)$.

Step 1: Surjectivity

We first show that the natural map $\Phi : \pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V) \to \pi_1(X)$ is surjective.

Let $[\gamma] \in \pi_1(X, x_0)$ be represented by a loop $\gamma : [0,1] \to X$. By compactness and the Lebesgue number lemma, we can partition $[0,1] = \bigcup_{i=0}^{n-1} [t_i, t_{i+1}]$ such that each $\gamma([t_i, t_{i+1}])$ lies entirely in $U$ or entirely in $V$.

Construct paths $\gamma_i : [t_i, t_{i+1}] \to X$ and connecting paths $\alpha_i$ from $x_0$ to $\gamma(t_i)$ in $U \cap V$ (using path-connectivity). The loop $\gamma$ is homotopic to the product: $\alpha_0 * \gamma_1 * \alpha_1^{-1} * \alpha_1 * \gamma_2 * \alpha_2^{-1} * \cdots * \alpha_{n-1} * \gamma_n * \alpha_0^{-1}$

Each term $\alpha_{i-1}^{-1} * \gamma_i * \alpha_i$ is a loop based at $x_0$ lying entirely in $U$ or $V$, hence in the image of $\Phi$. Therefore $[\gamma]$ is in the image, proving surjectivity.

Step 2: Defining the inverse

To show injectivity (and construct an explicit inverse), we use the universal property. Suppose $[\gamma] \in \pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V)$ maps to the trivial element in $\pi_1(X)$.

This means the corresponding loop in $X$ is nullhomotopic via some $H : [0,1] \times [0,1] \to X$ with $H(s, 0) = \gamma(s)$, $H(s, 1) = x_0$, and $H(0, t) = H(1, t) = x_0$.

Step 3: Refining the homotopy

Subdivide $[0,1] \times [0,1]$ into small squares such that each square maps entirely into $U$ or $V$. This uses compactness and the Lebesgue number lemma again.

For each square:

• If it maps to $U$, the corresponding portion of $\gamma$ can be contracted in $U$
• If it maps to $V$, the corresponding portion contracts in $V$
• Squares mapping to $U \cap V$ handle transitions

Step 4: Algebraic rewriting

Track how the subdivision allows us to rewrite $[\gamma]$ in the amalgamated product. Each square contributes relations showing that adjacent pieces (one in $U$, one in $V$) agree on their overlap in $U \cap V$.

The homotopy $H$ provides a sequence of moves in the amalgamated product: $[\gamma] = g_1 h_1 g_2 h_2 \cdots g_n h_n$ where $g_i \in \pi_1(U)$ and $h_i \in \pi_1(V)$. The contractibility in $X$ translates to relations showing this product equals the identity in the amalgamated product.

Step 5: Conclusion

Since $\Phi$ is surjective and injective, it is an isomorphism. The construction is natural in the sense that it commutes with continuous maps, confirming functoriality. ∎