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First homology of $F_k$
For this question I’m really not even sure where to start:
The fundamental group of the graph $F_k$ is isomorphic to the symmetric group $S_k$. Assume $k>1$, and consider the action of the graph on itself by the symmetric group. If $H_1(F_k;\mathbb{Z})$ denotes the first homology of $F_k$, then show that $H_1(F_k;\mathbb{Z})$ is the quotient of $\mathbb{Z}[S_k]$ by the ideal generated by $\sum_{i=1}^k [i]$, where $[i]$ is the image of a transposition $(i,i+1)$ under the map to $F_k$.
This is an exercise from Hatcher. I’m not even sure how to start, but it’s been on my mind for a while.
Thank you for any help,
A:
We are dealing with a semidirect product $K\rtimes S_k$ where the action of $K$ on $
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