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?