Notes:Homology/Real projective plane

[ilmath]\mathbb{RP}^2_A[/ilmath]

[ilmath]\xymatrix{ v\ \bullet \ar@/^.75pc/@{<-}[rr]^a \ar@/_.75pc/[rr]_a & & \bullet\ v }[/ilmath] Try and consider this as a solid disk called [ilmath]P[/ilmath] orientated clockwise (so the boundary of [ilmath]P[/ilmath] is [ilmath]a+a[/ilmath])
Set up [ilmath]A[/ilmath], denoted [ilmath]\mathbb{RP}^2_A[/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\rangle\cong\mathbb{Z}^1 & \langle v\rangle\cong\mathbb{Z}^1 }[/ilmath]

with:

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

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On paper I ended up with:

1. [ilmath]H_0\cong\mathbb{Z}[/ilmath]
2. [ilmath]H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} }[/ilmath]
3. [ilmath]H_2\cong 0[/ilmath]

[ilmath]\mathbb{RP}^2_B[/ilmath]

[ilmath]\xymatrix{ w\ \bullet \ar@{<-}[rr]^a \ar@<.8ex>[d]_b & & \bullet\ v \ar@<-.8ex>@{<-}[d]^b \\ v\ \bullet \ar[rr]_a & & \bullet\ w}[/ilmath] Text
Set up [ilmath]B[/ilmath], denoted [ilmath]\mathbb{RP}^2_B[/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,w\rangle\cong\mathbb{Z}^2 }[/ilmath]

With:

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

On paper I ended up with:

1. [ilmath]H_2\cong 0[/ilmath]
2. [ilmath]H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} }[/ilmath]
3. [ilmath]H_0\cong\mathbb{Z}[/ilmath]