# Van Kampen counterexamples

The van Kampen theorem states that if I have a space $$X = \cup_\alpha A_\alpha$$ where $$A_\alpha$$ are all path connected and if for any given $$\alpha, \beta, \gamma$$ we get $$A_\alpha \cap A_\beta \cap A_\gamma$$ is also path connected then the kernel of $$\Phi:*\Pi_1(A_\alpha) \to \Pi_1(X)$$ is generated by loops in the intersection of two to $$A$$s and not cancelling with its inverse. That is to say objects of the following form: suppose we have some loop $$\gamma$$ in $$A_\alpha \cap A_\beta$$ then it shows up in $$\Pi_1(A_\alpha)$$ as $$\gamma_\alpha$$. It also shows up in $$\Pi_1(A_\beta)$$ as $$\gamma_\beta$$ and the inverse of this loop is in $$\Pi_1(A_\beta)$$ as $$\gamma^{-1}_\beta$$.

The statement of the theorem is then that $$\ker\Phi$$ is generated by objects of the form $$\gamma_\alpha \gamma^{-1}_\beta$$

My question is if I relax $$A_\alpha \cap A_\beta \cap A_\gamma$$ being path connected and just demand that any pair $$A_\alpha \cap A_\beta$$ be path connected, what else can I get in the kernel?

• This doesn't answer your question, but it is worth noting that the map $\Phi$ need not be surjective in the case you describe. To see this, consider $X = S^1$ and three appropriately chosen arcs. In this case, the kernel is trivial. Nov 3, 2019 at 1:11

However there is a method of dealing with this which I published in 1967, namely to use many base points and so use the fundamental groupoid $$\pi_1(B,S)$$ on a set $$S$$ chosen according to the geometry of the situation: there is discussion of this idea in this mathoverflow entry. The idea was thought of in order to get a theorem which would compute at the same time also the fundamental group of the circle, a rather important example in topology, as well as a myriad of other cases. Why not?
The most general theorem of this type seems to be the following, from this paper, of 1984. Notice that for convenience we write also $$\pi_1(U,S)$$ for $$\pi_1(B, U \cap S)$$.
Theorem Let $$(B_\lambda)_{\lambda\in \Lambda}$$ be a family of subspaces of $$B$$ such that the interiors of the sets $$B_\lambda$$ $$(\lambda\in \Lambda)$$ cover $$B$$, and let $$S$$ be a subset of $$B$$. Suppose $$S$$ meets each path-component of each one-fold, two-fold, and each three-fold intersection of distinct members of the family $$(B_\lambda)_{\lambda\in \Lambda}$$. Then there is a coequalizer diagram in the category of groupoids: $$\bigsqcup_{\lambda,\mu\in\Lambda}\pi_1(U_\lambda\cap U_\mu, S)\rightrightarrows^\alpha_\beta \bigsqcup _{\lambda\in \Lambda}\pi_1(U_ \lambda,S)\to^\gamma\pi_1(B,S)$$ in which $$\bigsqcup$$ stands for the coproduct (disjoint union) in the category of groupoids, and $$\alpha$$, $$\beta$$, and $$\gamma$$ are determined by the inclusion maps $$U_\lambda\cap U_\mu\to U_\lambda$$, $$U_\lambda\cap U_\mu\to U_\mu$$, and $$U_\lambda\to B$$, respectively.