A closed connected manifold has top integral homology $\ZZ$ exactly when it is orientable.Suggest a slogan
Let \(M\) be a closed connected \(n\)-manifold. Then \[ H_n(M;\ZZ) \cong \begin{cases} \ZZ & M \text{ orientable},\\ 0 & \text{otherwise}, \end{cases} \qquad H_i(M;\ZZ) = 0 \text{ for } i > n \] [@Hat02], and the torsion subgroup of \(H_{n-1}(M;\ZZ)\) is trivial if \(M\) is orientable and \(\ZZ/2\) otherwise [@Hat02].