Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 20F / b

Use the orientationsThe simplicial chain complex haswithThe map ${\partial }_2$ is injective, so $H_2(K)=0$. The complex is connected, so $H_0(K)\cong \mathbb{Z}$ and $\mathrm{rank}{\partial }_1=3$. Hence $\ker {\partial }_1$ has rank $5-3=2$; quotienting by the rank-one primitive subgroup $\mathrm{im}{\partial }_2$ givesThere are no higher simplices. ThereforeThe result agrees with the Euler characteristic $4-5+1=0$.