Notes:Homology/Real projective plane
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Revision as of 01:48, 16 October 2016 by Alec (Talk | contribs) (Weird. Overflow now fixed. Guess I can't use xymatrix in a span tag!)
[ilmath]\mathbb{RP}^2_A[/ilmath]
The chain complexes are:with:
- [ilmath]\partial_2(P)=a+a=2a[/ilmath]
- [ilmath]\partial_1(a)=v-v=0[/ilmath]
- [ilmath]\partial_0(v)=0[/ilmath]
On paper I ended up with:
- [ilmath]H_0\cong\mathbb{Z}[/ilmath]
- [ilmath]H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} }[/ilmath]
- [ilmath]H_2\cong 0[/ilmath]
[ilmath]\mathbb{RP}^2_B[/ilmath]
The chain complexes are:With:
- [ilmath]\partial_2(P)=2(a+b)[/ilmath]
- [ilmath]\partial_1[/ilmath]
- [ilmath]\partial_1(a)=w-v[/ilmath]
- [ilmath]\partial_1(b)=v-w[/ilmath]
- [ilmath]\partial_0[/ilmath]
- [ilmath]\partial_0(v)=0[/ilmath]
- [ilmath]\partial_0(w)=0[/ilmath]
On paper I ended up with:
- [ilmath]H_2\cong 0[/ilmath]
- [ilmath]H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} }[/ilmath]
- [ilmath]H_0\cong\mathbb{Z}[/ilmath]