Notes:Homology/Torus

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

Situation

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

The chain complexes are:

[ilmath]\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 }[/ilmath]

With:

1. [ilmath]\partial_2(P)=a-b-a+b=0[/ilmath]
2. [ilmath]\partial_1[/ilmath]
   • [ilmath]\partial_1(a)=v-v=0[/ilmath]
   • [ilmath]\partial_1(b)=v-v=0[/ilmath]
3. [ilmath]\partial_0(v)=0[/ilmath]

On paper I ended up with:

• [ilmath]H_2\cong\mathbb{Z}[/ilmath]
• [ilmath]H_1\cong\mathbb{Z}^2[/ilmath]
• [ilmath]H_0\cong\mathbb{Z}[/ilmath]