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 we get such a map $\alpha$. What is the justification here?

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?

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 $X' = X \wedge S^2$ 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 we get such a map $\alpha$. What is the justification here?

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 we get such a map $\alpha$. What is the justification here?

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 $X' = X \wedge S^2$ 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 we get such a map $\alpha$. What is the justification here?

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 $X' = X \wedge S^2$ 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 we get such a map $\alpha$. What is the justification here?