From https://mathoverflow.net/questions/100155/second-homotopy-group-of-cayley-complex-- for $<X,R>$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules

$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\rightarrow\mathbb{Z}G^{\oplus X}\rightarrow \mathbb{Z}G\rightarrow \mathbb{Z}\rightarrow 0$$ .

I dont have any idea from where it came, neither I dont have any theoritical knowledge of such a thing. Can anybody provide me some reference?It will of great help!

From this Q&A -- for $\langle X,R\rangle$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules

$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\rightarrow\mathbb{Z}G^{\oplus X}\rightarrow \mathbb{Z}G\rightarrow \mathbb{Z}\rightarrow 0.$$

I don't have any idea from where it came, neither I dont have any theoretical knowledge of such a thing. Can anybody provide me some reference? It will be of great help!

From https://mathoverflow.net/questions/100155/second-homotopy-group-of-cayley-complex-- for $$<X,R>$$ a finite presentation of a group $G$, there is an exact sequence of $$\mathbb{Z}G$$ modules

$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\rightarrow\mathbb{Z}G^{\oplus X}\rightarrow \mathbb{Z}G\rightarrow \mathbb{Z}\rightarrow 0$$ .

I dont have any idea from where it came, neither I dont have any theoritical knowledge of such a thing. Can anybody provide me some reference?It will of great help!

From https://mathoverflow.net/questions/100155/second-homotopy-group-of-cayley-complex-- for $<X,R>$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules

$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\rightarrow\mathbb{Z}G^{\oplus X}\rightarrow \mathbb{Z}G\rightarrow \mathbb{Z}\rightarrow 0$$ .

I dont have any idea from where it came, neither I dont have any theoritical knowledge of such a thing. Can anybody provide me some reference?It will of great help!