3-manifold’s homology

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

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


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

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


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


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



  • 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)

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