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I am reading Theorem $2.1$ of the paper The Volume and Chern-Simons Invariant of a Representation. I ran into a problem in proving the last part of it, i.e., to prove that the cokernel of the map $F_2 \to F_1$ is $K$.

Here I am trying to describe the problem in a simple form, and I think anyone can understand it without going through the proof of Theorem $2.1$.

If $X$ is any set, then denote $\mathbb{Z}[X]$ to be the free abelian group generated over the set $X$. Let $G$ be a group and $H$ its subgroup and define a group homomorphism $\rho: \mathbb{Z}[G^2] \to \mathbb{Z}[G/H]$ as $\rho(n(g_1,g_2))= nHg_2 -nHg_1.$ What is the $\ker(\rho)$?

As per the requirement of the proof of Theorem $2.1$ and if my calculations are correct then the kernel must be generated by the following set $\{ (hg,g) - (g_2g,g) +(g_1g,g) -(g_1 g_2^{-1}g,g)~|~ g_1,g_2,g \in G, ~h\in H\}.$

I don't know why $Hg_2g-Hg_1g-Hg + Hg_1g_2 ^{-1}g=0$.

Can someone help me in proving this?

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  • $\begingroup$ Is H normal in G or is G/H simply the set of cosets? $\endgroup$
    – user622002
    Commented Feb 6, 2020 at 10:01
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    $\begingroup$ Your generating seems weird to me. First $g$ is not on the list of variables on the right, but that is not very important. More to the point, when $m'=0$ and $m=1$, the element is very clearly not in the kernel. Also, what is the point of $h_1$ and $h_2$ when obviously they could be replaced by any $h\in H$, or even just by $1$? $\endgroup$ Commented Feb 6, 2020 at 10:18
  • $\begingroup$ user622002: $G/H$ is simply the set of cosets. $H$ need not be normal. $\endgroup$
    – eyp
    Commented Feb 6, 2020 at 10:26
  • $\begingroup$ Lama: As per your observations, I have made changes to the question. Moreover, the point that taking $m=1, m'=0$ is right. Let me check my calculations. And I think taking $h=1$ will change the generating set. $\endgroup$
    – eyp
    Commented Feb 6, 2020 at 10:33
  • $\begingroup$ Lama: I have asked a question related to this question at math.stackexchange.com/questions/3536583/…. $\endgroup$
    – eyp
    Commented Feb 6, 2020 at 12:04

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