What is the fundamental group of a wedge sum in general? e.g. including the times when van Kampen cannot help us.
The Wikipedia article on wedge sums mentions that
Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) under which the fundamental group of the wedge sum of two spaces $X$ and $Y$ is the free product of the fundamental groups of $X$ and $Y$.
This seems strangely worded. The conditions on the space for van Kampen are that it is
- the union of path-connected open sets $A_\alpha$,
- each contains the basepoint,
- each intersection $A_\alpha\cap A_\beta$ is path-connected,
- each intersection $A_\alpha\cap A_\beta\cap A_\gamma$ is path-connected;
then it follows that the the kernel of $\Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}$ for $\omega\in\pi_1(A_\alpha\cap A_\beta)$, and hence $\Phi$ induces an isomorphism $\pi_1(X)\cong \ast_\alpha\pi_1(A_\alpha)/N$.
(I fix a basepoint for the rest of this question, all wedge sums are identified at this basepoint.)
Back to wedge sums. If we are looking at the wedge sum $\bigvee_\alpha A_\alpha$ where each $A_\alpha$ is path connected, then these conditions are obviously true. And since the intersection of any $A_\alpha$ is trivial, the fundamental group of the wedge sum is the free product of the groups.
The only problem that I can see arising is when $A_\alpha$ is not path connected. The fundamental group of a disconnected space $X\cup Y$ is $\pi_1(X)$ where the basepoint falls in $X$ (correct me if I am wrong, maybe I have over-simplified this). What is the fundamental group of this wedge sum, then? I would guess that it is $$\ast_\alpha\pi_1(A_\alpha^\prime),$$ where $A_\alpha^\prime$ is the component of $A_\alpha$ containing the basepoint. If so, then surely $\pi_1(A_\alpha)=\pi_1(A_\alpha^\prime)$, and there is no problem after all, the fundamental group is just the free product of each fundamental group. Is this correct (the fundamental group is always $\pi_1(\bigvee A_\alpha)=\ast\pi_1(A_\alpha)$)?
The simple example I scrawled on my piece of paper is below. I figure the fundamental group is $$\pi_1(\mathsf{Green}\vee\mathsf{Blue}\vee\mathsf{Red})=\Bbb{Z}\ast\Bbb{Z}\ast C_1=\Bbb{Z}\ast\Bbb{Z},$$ which makes sense, no loop at the basepoint can be in the leftmost or rightmost components, they must be (homotopic to a loop) in the figure eight graph.
P.S. a side question, why do we use \vee
for the wedge sum rather than \wedge
?
Thanks in advance :-)