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In hatcher's algebraic topology (theorem 1.20)

If $X$ is the union of path-connected open sets $A_\alpha$ each containing the basepoint $x_0 \in X$ and if each intersection $A_\alpha \cap A_\beta$ is path-connected, then the homomorphism $\Phi : *_\alpha \pi_1(A_\alpha) \to \pi_1(X)$ is surjective. If in addition each intersection $A_\alpha \cap A_\beta \cap A_\gamma$ is path-connected, then the kernel of $\Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}$

Here $i_{\alpha\beta}$ and $i_{\beta\alpha}$ are the homomorphisms induced by the inclusions $\pi_(A_\alpha \cap A_\beta) \hookrightarrow \pi_1(A_\alpha)$ and $\pi_1(A_\alpha \cap A_\beta) \hookrightarrow \pi_1(A_\beta)$ respectively.

I'm having trouble visualizing what $i_{\alpha \beta}$ and $i_{\beta \alpha}$ looks like. In many problems I've looked at this doesn't matter since $N = 0$, but are there examples where they give $N \neq 0$?

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    $\begingroup$ Take time to spell names properly with capital letters (van Kampen, Hatcher). $\endgroup$
    – Paul Frost
    Sep 9, 2022 at 23:02

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Here is an example with $N \neq 0$. Take a genus two surface S, viewed as a connect sum of two tori. Then the decomposition of S along the circle on which the connect sum is done, gives an example.

Note that the maps $i_{\alpha \beta}, i_{\beta \alpha}$ are induced by continuous maps of topological spaces, so they can be viewed in the usual way, i.e. a loop $\varphi$ goes to a loop $f \circ \varphi$

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