# a question about van kampen theorem

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 $$N \neq 0$$?

• Take time to spell names properly with capital letters (van Kampen, Hatcher). Sep 9, 2022 at 23:02

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 $$f \circ \varphi$$