Section 4.4 The Seifert–Van Kampen Theorem

Patching coverings.

Suppose \(X = U \cup V\) with \(U, V\) open. A covering of \(X\) restricts to coverings on \(U\) and \(V\) that are isomorphic over \(U \cap V\text{.}\) Conversely, given coverings \(p_U \colon Y_1 \to U\) and \(p_V \colon Y_2 \to V\) with an isomorphism

\begin{equation*} \theta \colon p_U^{-1}(U \cap V) \xrightarrow{\ \cong\ } p_V^{-1}(U \cap V) \end{equation*}

of coverings over \(U \cap V\text{,}\) one can patch them to obtain a covering \(p \colon Y \to X\) together with isomorphisms \(\varphi_U \colon Y_1 \cong p^{-1}(U)\) and \(\varphi_V \colon Y_2 \cong p^{-1}(V)\) satisfying \(\theta = \varphi_V^{-1} \circ \varphi_U\) over \(U \cap V\text{.}\)

Theorem 4.19. Seifert–Van Kampen.

Let \(X = U \cup V\) be locally path-connected and semilocally simply connected, with \(U\text{,}\) \(V\) open and path-connected and \(U \cap V\) path-connected. Let \(x_0 \in U \cap V\) and write \(i_{U,*} \colon \pi_1(U \cap V, x_0) \to \pi_1(U, x_0)\) and \(i_{V,*} \colon \pi_1(U \cap V, x_0) \to \pi_1(V, x_0)\) for the maps induced by the inclusions. Then the diagram

\begin{equation*} \begin{array}{ccc} \pi_1(U \cap V,\, x_0) & \xrightarrow{\ i_{V,*}\ } & \pi_1(V,\, x_0) \\[6pt] \downarrow\,{i_{U,*}} & & \downarrow\,{j_{V,*}} \\[6pt] \pi_1(U,\, x_0) & \xrightarrow{\ j_{U,*}\ } & \pi_1(X,\, x_0) \end{array} \end{equation*}

is a pushout in the category of groups: given any group \(G\) and homomorphisms \(h_U \colon \pi_1(U, x_0) \to G\) and \(h_V \colon \pi_1(V, x_0) \to G\) with \(h_U \circ i_{U,*} = h_V \circ i_{V,*}\text{,}\) there exists a unique homomorphism \(h \colon \pi_1(X, x_0) \to G\) with \(h \circ j_{U,*} = h_U\) and \(h \circ j_{V,*} = h_V\text{.}\)

Proof.

Via Proposition 4.17, \(h_U\) and \(h_V\) determine based \(G\)-coverings \(P_U \colon \tilde{Y}_U \to U\) and \(P_V \colon \tilde{Y}_V \to V\text{,}\) with basepoints \(y_U,y_V\) over \(x_0\text{.}\)

The condition \(h_U \circ i_{U,*} = h_V \circ i_{V,*}\) means that the restrictions to \(U \cap V\) are isomorphic as based \(G\)-coverings. Since \(U \cap V\) is path-connected, Lemma 2.8 ensures this isomorphism is unique (it is determined by sending \(y_U\) to \(y_V\)). Patching \(\tilde{Y}_U\) and \(\tilde{Y}_V\) along this isomorphism gives a \(G\)-covering \(p \colon Y \to X\text{,}\) which under the bijection of Proposition 4.17 corresponds to the unique homomorphism \(h \colon \pi_1(X, x_0) \to G\text{.}\)

Example 4.20. The figure eight.

Let \(X=S^1\vee S^1\) be the figure eight, with wedge point \(x_0\text{.}\) Choose a point \(q_L\) on the left-hand circle and a point \(q_R\) on the right-hand circle, both different from \(x_0\text{,}\) and set

\begin{equation*} U=X\setminus\{q_R\}, \qquad V=X\setminus\{q_L\}. \end{equation*}

These are open, path-connected subspaces of \(X\text{.}\) Collapsing the punctured circle shows that \(U\) deformation retracts onto the left-hand circle and \(V\) deformation retracts onto the right-hand circle. Hence

\begin{equation*} \pi_1(U,x_0)\cong\mathbb Z=\langle a\rangle, \qquad \pi_1(V,x_0)\cong\mathbb Z=\langle b\rangle. \end{equation*}

The intersection \(U\cap V\) is a tree, so it is contractible. The maps from its trivial fundamental group impose no relations, and Theorem 4.19 gives

\begin{equation*} \pi_1(S^1\vee S^1,x_0) \cong \mathbb Z*\mathbb Z \cong F(a,b), \end{equation*}

the free group on the two loops. Thus the figure eight is the first basic example where the fundamental group is nonabelian.

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