Let \(G\) be an abelian group and \(n\geq 1\). A Moore space \(M(G, n)\) is a CW complex \(X\) with \[ \tilde H_i(X;\ZZ)\cong \begin{cases} G & i = n, \\ 0 & i\neq n. \end{cases} \]
For every abelian group \(G\) and \(n\geq 1\) a Moore space \(M(G, n)\) exists. Choose a free resolution \(0\to F_1\mapsvia{\varphi} F_0\to G\to 0\) with a basis \((a_\alpha)_{\alpha}\) of the free abelian group \(F_0\) and a basis \((b_\beta)_{\beta}\) of the free abelian group \(F_1\); such a resolution exists because every subgroup of a free abelian group is free. Let \(X^n\coloneqq\bigvee_\alpha S^n_\alpha\) be a wedge of \(n\)-spheres, one for each \(a_\alpha\), and identify \(H_n(X^n;\ZZ)\) with \(F_0\) by sending the class of \(S^n_\alpha\) to \(a_\alpha\). For each \(\beta\), attach an \((n+1)\)-cell \(e^{n+1}_\beta\) along a map \(f_\beta\colon S^n\to X^n\) with \((f_\beta)_*[S^n] = \varphi(b_\beta)\), which exists because the Hurewicz map \(\pi_n(X^n)\to H_n(X^n;\ZZ)\) is surjective; composed with the collapse \(X^n\to S^n_\alpha\), the map \(f_\beta\) has degree equal to the \(a_\alpha\)-coordinate of \(\varphi(b_\beta)\). The resulting CW complex \(X\) has cellular chain complex \(0\to F_1\mapsvia{\varphi}F_0\to 0\) in degrees \(n+1\) and \(n\), so \(\tilde H_n(X;\ZZ)\cong\coker\varphi\cong G\) and \(\tilde H_{n+1}(X;\ZZ)\cong\ker\varphi = 0\) [@Hat02].
The sphere \(S^n\) is an \(M(\ZZ, n)\). For \(m\geq 2\), the space \(S^n\cup_m e^{n+1}\) obtained by attaching an \((n+1)\)-cell to \(S^n\) along a map \(S^n\to S^n\) of degree \(m\) is an \(M(\ZZ/m, n)\).
See [@Hat02].