For closed connected 3-manifolds, orientability determines the top two homology groups from $H_1$.Suggest a slogan
Let \(M\) be a closed connected \(3\)-manifold and write \(H_1(M;\ZZ) \cong \ZZ^r \oplus F\) with \(F\) finite. Then \[ H_*(M;\ZZ) \cong \begin{cases} \qty{ \ZZ,\; \ZZ^r \oplus F,\; \ZZ^r,\; \ZZ } & M \text{ orientable},\\ \qty{ \ZZ,\; \ZZ^r \oplus F,\; \ZZ^{r-1} \oplus \ZZ/2,\; 0 } & M \text{ nonorientable}, \end{cases} \] in degrees \(0, 1, 2, 3\), with all higher groups zero; in the nonorientable case \(r \geq 1\) [@Hat02].
\(H_1\) can have torsion: the lens space \(L_p(1,1)\) has \(H_1\cong\ZZ/p\).