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)

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s