I am reading a paper which makes the following claim:


let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$.
Let $X' = X \vee S^2$ be the wedge sum of $X$ with the sphere.
Then $X'$ induces a map $\alpha: S^2 \to X'$ which generates a free copy of $\mathbb{Z}[G] = \mathbb{Z}[\pi\_1(X')]$ inside $\pi\_2(X')$.
Here, $\mathbb{Z}[G]$ denotes the group ring of $G$ over the integers.


I don't follow with how $\alpha$ includes $\mathbb{Z}[G]$ in $\pi\_2(X')$. What is the justification here?



Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group
Let $\tilde Y$ denote the universal cover of $Y$. Then
$\tilde{X'}$ is formed from $\tilde X$ by gluing a copy of $S^2$ to each preimage of the basepoint of $X$. Then by the Hurewicz isomorphism, we have
$$ \pi\_2(X') = \pi\_2(\tilde{X'}) = H\_2(\tilde X) \oplus \bigoplus\_{g \in \pi\_1(X)} \mathbb Z = \pi\_2(X) \oplus \mathbb Z[\pi\_1(X)],$$
since the classes of the glued spheres are independent and generate a summand of $H\_2(\tilde X)$, and are freely permuted by $\pi\_1(X)$.


