I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $\beta$
$\beta_i=\dim(Z_i)-\dim(B_i)$ where $Z_i=\ker(\partial_i)$ and $B_i$=im$(\partial_i{+}_1))$
I have $\beta_1=\dim(Z_1)-\dim(B_1)=2-3$, am I going about this the right way?