Form the triangular parachute $P$ from a regular $2$-simplex with all vertices identified to a single vertex. Label the vertex $v$, the edges $a,b,c$, and the face $f$. We wish to determine the simplicial homology groups. We have $\Delta_0 = \la v \ra, \Delta_1 = \la a,b,c \ra$ and $\Delta_2 = \la f \ra$ (of course, all brackets in this context mean the free abelian group on the enclosed generators).

For $\delta_0$, we have $\im \delta_0 = \set{0}$ and $\ker \delta_0 = \Delta_0$. For $\delta_1$, we have $\delta_1(a) = \delta_1(b) = \delta_1(c) = v-v=0$. Thus, $\im \delta_1 = \set{0}$ and $\ker \delta_1 = \Delta_1$.

For $\delta_2$, we have $\delta_2(f) = a-b+c$. Then, $\ker \delta_2 = \set{0}$, and $\im \delta_2$ is precisely $I_2 = \setbuilder{ka - kb + kc}{k \in \mathbb{Z}}$.

$\delta_3$ and higher are all the zero map.

It's clear that we have $H_0(P) = \la v \ra / \set{0} = \la v \ra \cong \mathbb{Z}$.

Next, $H_1(P) = \la a,b,c \ra / I_2$. Consider the homomorphism $\varphi : \la a,b,c \ra \rightarrow \mathbb{Z}^2$ given by $\varphi(ra + sb + tc) = (r-t, s+t)$. Observe that $\ker \varphi$ is precisely $\setbuilder{ra + sb + tc}{s=-r, t=r}$, which is exactly an equivalent description of $I_2$. We conclude $H_1(P) \cong \im \varphi = \mathbb{Z}^2$.

$H_2(P)$ is $\set{0}$ because $\delta_2$ is injective, and all higher homology groups are zero as well.