Moore space (algebraic topology)

For an abelian group $G$ and an integer $m\ge 1$, a Moore space $M(G,m)$ is a CW complex whose reduced homology is $G$ in degree $m$ and zero in every other degree.

Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.

Attaching a two-cell to $S^1$ by a map of degree $n$ gives the Moore space $M(\mathbb{Z}/n,1)$. Its positive-dimensional cellular chain complex isso $H_1\cong \mathbb{Z}/n$ and all its other reduced integral homology groups vanish.