I was wondering how one can compute the fundamental group of the wedge sum of a sphere and 2 circles , i know the fundamental group is Z*Z ,and that the fundamental group of a wedge sum is the free product of each fundamental group but is it possible to get it by Van Kampen ? I couldn't do it .Any help is appreciated .
[Math] The fundamental group of some wedge sum
algebraic-topology
Best Answer
The two techniques you mentioned in your question: "the fundamental group of a wedge sum is the free product of each fundamental group" and "getting it by Van Kampen" are the same. Let $X$ and $Y$ be any nice spaces (we'll specify what nice means below) and let $S$ be their wedge sum at a common point, which we'll call $p$.
Then the copy of $X$ in $S$ admits an open neighborhood $U_X$ which is the union of the copy of $X$ and a small open neighborhood of $p$ in $Y$ which deformation retacts to $p$ (so one reasonable definition of "nice" would be "each point has a contractible neighborhood," although this can be relaxed -- but at least it certainly applies in your example). Define $U_X$ similarly. Since $U_X \cap U_Y$ is contractible, you recover "the fundamental group of a wedge sum is the free product of each fundamental group" from Van Kampen's theorem.