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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.

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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 $x \in U$.

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  • $\begingroup$ My bad, I had a wrong definition of a contractible. I thought it meant that $U \hookrightarrow X$ was nullhomotopic, not necessarily $U \rightarrow U$. $\endgroup$ Apr 10, 2020 at 15:22
  • $\begingroup$ @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". $\endgroup$ Apr 10, 2020 at 15:23

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