# 3-manifold’s homology

its $H_*(M;\mathbb{Z})$ is:

• $w_1(M)\neq 0$ but $\partial M=\emptyset$

$H_0(M)=\mathbb{Z}$

$H_1(M)={\mathbb{Z}}^r\oplus T$

$H_2(M)={\mathbb{Z}}^{r-1}\oplus {\mathbb{Z}}_2$

$H_3(M)=0$

• For $w_1(M)=0$ and $\partial M=\emptyset$

$H_0(M)=\mathbb{Z}$

$H_1(M)={\mathbb{Z}}^r\oplus T$

$H_2(M)={\mathbb{Z}}^r$

$H_3(M)={\mathbb{Z}}$

• For $w_1(M)=0$ and $\partial M\neq\emptyset$

$H^0(M)\cong H_3(M,\partial M)\qquad H_0(M)\cong H^3(M,\partial M)$

$H^1(M)\cong H_2(M,\partial M)\qquad H_1(M)\cong H^2(M,\partial M)$

$H^3(M)\cong H_0(M,\partial M)\qquad H_2(M)\cong H^1(M,\partial M)$

• For $w_1(M)=0$ we have $b_1={\rm rank}H_1(M)\ge\#(\mbox{handles of }$ $\partial M)$