Seifert-Van Kampen Theorem

Statement

Let $X$ be a topological space such that it can be decomposed into open sets $X={\bigcup }_{\alpha }{U}_{\alpha }$ such that

• each open set contains the basepoint $x_0\in U_{\alpha }$
• $U_{\alpha }$ is open and path-connected
• $U_{\alpha }\cap U_{\beta }$ is path-connected
• $U_{\alpha }\cap U_{\beta }\cap U_{\gamma }$ is path-connected

Then

{\prod_\alpha}^*\pi_1(U_\alpha) \bigg/ \langle {j_{\beta}}_* [\gamma]{j_{\alpha}}_*[\gamma]^-1 \ | \ \gamma \in \pi_1(U_\alpha \cap U_\beta) \,\,\, \forall \alpha, \beta\rangle$$ Note ${j_\alpha}_*$ is the pushforward along the inclusion as seen in the diagram below. ```tikz \usepackage{tikz-cd} \tikzcdset{scale cd/.style={every label/.append style={scale=#1}, cells={nodes={scale=#1}}}} \begin{document} \begin{tikzcd}[scale cd=2, sep=huge] & U_\alpha \arrow[dr, hookrightarrow, "i_\alpha"]&\\ U_\alpha \cap U_\beta \arrow[ur, hookrightarrow, "j_\alpha"]\arrow[dr, hookrightarrow, "j_\beta"] & &X\\ & U_\beta \arrow[ur, hookrightarrow, "i_\beta"] & \end{tikzcd} \end{document} ``` # Proof #todo # Examples ## Wedge sum Let $I$ be a set (e.g. $\{1,2\}$). Let $$ \bigvee_I S^1 = \bigsqcup_{i\in I}S^1_i \bigg/ \begin{array}1 \text{glue together all points }\\(0,1) \ \text{on each }S^1\end{array}$$ This is called the _wedge sum_. Let $U_i = S_i^1 \bigcup_j \text{small open set of each }(1,0)\in S^1_j$. $U_i \cap U_j$ are path connected and $\pi_1 \simeq 0$, since they are a bunch of curved lines connected in the "middle" at a point they are contractible. Thus, $$\pi_1\left(\bigvee S^1\right) = {\prod_{s \in I}}^*\mathbb{Z}\cdot s$$ ## Torus #todo # References [[@hatcher2002]] - chapter 1.2, theorem 1.20.