Notes:Homology/Torus

We denote the torus as $\mathbb{T}$ here. This may not be the site convention. (see: ASN).

Situation

$\xymatrix{ v\ \bullet \ar@{<-}[rr]^a \ar@<.8ex>@{<-}[d]_b & & \bullet\ v \ar@<-.8ex>@{<-}[d]^b \\ v\ \bullet & & \bullet\ v \ar[ll]^a }$ Text
Set up for $\mathbb{T}$

The chain complexes are:

$\xymatrix{ 0 \ar[r]^{\partial_3} & C_2 \ar[r]^{\partial_2} \ar@2{->}[d] & C_1 \ar[r]^{\partial_1} \ar@2{->}[d] & C_0 \ar[r]^{\partial_0=0} \ar@2{->}[d] & 0 \\ & \langle P\rangle\cong\mathbb{Z}^1 & \langle a,b\rangle\cong\mathbb{Z}^2 & \langle v\rangle\cong\mathbb{Z}^1 }$

With:

1. $\partial_2(P)=a-b-a+b=0$
2. $\partial_1$
   • $\partial_1(a)=v-v=0$
   • $\partial_1(b)=v-v=0$
3. $\partial_0(v)=0$

On paper I ended up with:

• $H_2\cong\mathbb{Z}$
• $H_1\cong\mathbb{Z}^2$
• $H_0\cong\mathbb{Z}$