# Fundamental group of wedge sum of CW complex

Consider two pointed CW complexes $$(X,x_0),(Y,y_0)$$ and their wedge sum $$(X \vee Y,a)$$ with $$a$$ the identification of the two base points. I want to give a sufficient condition under which we have

$$\pi_1(X \vee Y, a) = \pi_1(X,x_0) \ast \pi_1(Y,y_0) ~~.$$

Right now it is not clear to me whether we even need any additional assumption or whether this is true for any CW complexes. In case we need more conditions I am also looking for a counterexample to the case where there are no conditions.

One way I am considering is Seifert-Van-Kampen. Then we only need that there exists some neighborhood $$U$$ of $$a \in X \vee Y$$ which has trivial fundamental group. Sufficient for that would be:

Both, $$X$$,$$Y$$ have a neighborhood of $$x_0$$, $$y_0$$ which (stronlgy) deformation retracts to that point.

I am not sure whether this is true for CW complexes, but it could very well be. It is true that CW complexes are locally contractible, but that is not enough:

First off, for a contractible neighborhood $$U$$ of $$x_0$$ we do not even have $$\pi_1(U,x) = 0$$ (see comments). Moreover, there are contractible pointed spaces (even such that strongly deformation retract to a point) s.t. their wedge sum is not contractible. An example can be found here. However, this space is not a CW complex, so I do still have hope that the theorem holds without any further assumptions.

Yes, any point $$x$$ in a CW complex has a neighborhood which deformations retracts onto $$x$$. This is part of Proposition A.4 in Hatcher's Algebraic Topology for example.
First off, for a contractible neighborhood $$U$$ of $$x_0$$ we do not even have $$\pi_1(U,x) = 0$$.
Actually we do...? A contractible space has trivial homotopy groups. Is the issue with base points? A contractible space is path-connected, and any path between two points can be used to define isomorphisms between the fundamental groups based at these two points. So if a (sub)space $$U$$ deformation retracts onto some $$x_0$$, then $$\pi_1(U,x) = 0$$ for any $$x \in U$$.
• My bad, I had a wrong definition of a contractible. I thought it meant that $U \hookrightarrow X$ was nullhomotopic, not necessarily $U \rightarrow U$. Apr 10, 2020 at 15:22
• @G.Chiusole I see. This property doesn't really have a name that I'm aware of, but I guess a space $X$ such that any point $x$ has a neighborhood $U$ with $U \to X$ nullhomotopic would be called "semi-locally contractible". Apr 10, 2020 at 15:23