Let \(M\) be a closed connected 3-manifold, and write \(H_1(M; \mathbb{Z})\) as \(\mathbb{Z}^r \oplus F\), the direct sum of a free abelian group of rank \(r\) and a finite group \(F\). Show that \(H_2(M; \mathbb{Z})\) is \(\mathbb{Z}^r\) if \(M\) is orientable and \(\mathbb{Z}^{r-1} \oplus \mathbb{Z}_2\) if \(M\) is nonorientable. In particular, \(r \geq 1\) when \(M\) is nonorientable. Using Exercise 6, construct examples showing there are no other restrictions on the homology groups of closed 3-manifolds.
Solution
Write \[ H_1(M;\mathbb Z)\cong\mathbb Z^r\oplus F \] with \(F\) finite.
If \(M\) is orientable, Poincaré duality and the universal coefficient theorem give \[ H_2(M;\mathbb Z)\cong H^1(M;\mathbb Z) \cong\operatorname{Hom}(H_1(M),\mathbb Z) \cong\mathbb Z^r. \]
Suppose \(M\) is nonorientable. Corollary 3.28 says that the torsion subgroup of \(H_2(M;\mathbb Z)\) is exactly \(\mathbb Z_2\). Also every closed odd-dimensional manifold has Euler characteristic zero. Since \(M\) is connected and nonorientable, \[ H_0(M)\cong\mathbb Z, \qquad H_3(M)=0. \] Hence \[ 0=\chi(M)=1-r+\operatorname{rank}H_2(M), \] so \[ \operatorname{rank}H_2(M)=r-1. \] Therefore \[ \boxed{H_2(M;\mathbb Z)\cong\mathbb Z^{r-1}\oplus\mathbb Z_2.} \] In particular \(r\ge1\).
It remains to see that there are no further restrictions. Write \[ F\cong\mathbb Z_{m_1}\oplus\cdots\oplus\mathbb Z_{m_s}. \] For orientable examples, take the connected sum of \(r\) copies of \(S^1\times S^2\) and lens spaces \(L(m_j,1)\). Exercise 6 gives \[ H_1\cong\mathbb Z^r\oplus F, \qquad H_2\cong\mathbb Z^r. \]
For nonorientable examples, let \(N\) be the twisted \(S^2\)-bundle over \(S^1\), obtained as the mapping torus of a reflection of \(S^2\). Its mapping-torus sequence gives \[ H_1(N)\cong\mathbb Z, \qquad H_2(N)\cong\mathbb Z_2. \] Taking the connected sum of \(N\), \(r-1\) copies of \(S^1\times S^2\), and the same lens spaces yields \[ H_1\cong\mathbb Z^r\oplus F, \qquad H_2\cong\mathbb Z^{r-1}\oplus\mathbb Z_2. \] Thus every group pattern allowed by the formulas occurs.