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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!

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This object is an algebraic invariant of the presentation 2-complex $Z$ of the finite presentation $\langle X\mid R\rangle$ of the group $G$ you are considering.

$Z$ is a finite 2-complex, and to obtain the above invariant you form the (augmented) cellular chain complex of the universal cover $\tilde{Z}$ of $Z$ with $\mathbb{Z}G$ action induced by the monodromy action of $\pi_1(Z)\cong G$ on $\tilde{Z}$.

$$…\to C_3(\tilde{Z})\to C_2(\tilde{Z})\to C_1(\tilde{Z})\to C_0(\tilde{Z})\to \mathbb{Z}\to 0$$

Now, as $Z$ has dimension 2 as CW-complex, $\tilde{Z}$ inherits the structure of a dim 2 CW-complex, so the free $\mathbb{Z}G$-modules $C_i(\tilde{Z})$ vanish for $i>2$. Note that this gives us the exact sequence you want but with the second homology module $ker(\partial_2: C_2( \tilde{Z})\to C_1(\tilde{Z}))=H_2(\tilde{Z})$ in place of $\pi_2(Z)$ (as each $C_i(\tilde{Z})$ for $i=0,1,2$ is a $\mathbb{Z}G$-module of rank the number of i-cells in $Z$, which is precisely $1,|X|, |R|$ respectively). But $H_2(\tilde{Z})$ and $\pi_2(Z)$ coincide by the fact that $\tilde{Z}$ is simply connected, so it’s second homology and homotopy modules are isomorphic by Hurewicz isomorphism, and then higher homotopy groups are invariant under covering spaces, so $\pi_2(\tilde{Z})\cong \pi_2(Z)$.

This object is a homotopy invariant of $Z$ (after fixing an identification of $G$ with $\pi_1(Z)$). A good resource for this topic is ‘2-dimensional homotopy theory and combinatorial group theory’, but unfortunately I don’t have an online copy I can link to. Please let me know if there are any points you would like me to expand on.

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  • $\begingroup$ Thanks for ur reply. But I have one doubt. I have read cellular chain complex (for calculating homology) but I couldn't understand monodromy action since as per i know monodromy action comes in homotopy (not homology as per my little knowledge). So would u please tell some references? It will be of great help. 2nd doubt is that-since universal cover $\tilde{X}$ is 2-complex then shouldn't $C_{3}(\tilde X)$ be zero ? (Since theres no 3-cell) $\endgroup$
    – Souvik Mandal
    Commented Dec 21, 2023 at 6:06
  • $\begingroup$ On your second point you are of course correct this was a typo, it should be $H_2= ker \delta_2$, will edit accordingly, the argument should still be correct. In regards to the monodromy action in homology, this also goes more generally by the name of ‘homology with local coefficients ‘, a reference in page 328 of Hatcher where they give general construction, which we apply with $M=\mathbb{Z}G$. $\endgroup$
    – William Thomas
    Commented Dec 21, 2023 at 10:13
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    $\begingroup$ In this case, the “monodromy action” isn’t very complicated. Just notice that $G$ acts freely on the cells of the universal cover. Choosing orbit representatives identifies the cells of the universal cover with a disjoint union of copies of $G$, by orbit-stabiliser. The chain groups are formal sums of cells, i.e. direct sums of copies of $\mathbb{Z}G$. $\endgroup$
    – HJRW
    Commented Dec 21, 2023 at 12:20

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