Im trying to solve a Problem from Hatcher: Compute the simplicial homology groups of the $\Delta$-complex obtained from n+1 simplices $\Delta_0^2,\Delta_1^2,...,\Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge ($=:e_0$) , and for $i>0$ identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$ of $\Delta_i^2$ to a single edge ($=:e_i$) and the edge $[v_0,v_2]$ to the edge $[v_0,v_1]$ of $\Delta_{i-1}^2$.
I can calculate that $\partial_2 \Delta_0^2 = e_0$ and for $i>0\ $ $\partial_2 \Delta_i^2 = 2e_i-e_{i-1}$ and $\partial_1 e_i =0$ which yields:
$H_1(X) \cong \frac{ker\partial_1}{Im\partial_2}\cong \frac{ \mathbb{Z}<e_0,e_1,...,e_n>}{<e_0,2e_1-e_0,2e_2-e_1,...,2e_n-e_{n-1}>}$
and i cant simplify this. The correct solution is supposedly $\mathbb{Z}_{2^n}$.
I tried to tinker around with lower dimensional solutions and while i understand $\frac{\mathbb{Z}<e_0,e_1>}{<e_0,2e_1-e_0>} \cong \mathbb{Z}_2$ by rewriting the denominator as $<e_0,2e_1-e_0>=\mathbb{Z}(1,0)\oplus\mathbb{Z}(-1,2)\cong \mathbb{Z}(1,0)\oplus\mathbb{Z}(0,2)= <e_0,2e_1>$, i cant make sence of it with three generators: $\frac{\mathbb{Z}<e_0,e_1,e_2>}{<e_0,2e_1-e_0,2e_2-e_1>}$.
Could somebody please explain the case of three generators?
I would be very thankful for any help!