Difference between revisions of "Notes:Homology/Real projective plane"
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(Saving work) |
(Fixed some typos, added "skeleton" of second "set up" for RP2) |
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| <center><span style="font-size:1.5em;"><m>\xymatrix{ v\ \bullet \ar@/^.75pc/@{<-}[rr]^a \ar@/_.75pc/[rr]_a & & \bullet\ v }</m></span><br/>Try and consider this as a solid disk called {{M|P}} orientated clockwise<br/>(so the boundary of {{M|P}} is {{M|a+a}})</center> | | <center><span style="font-size:1.5em;"><m>\xymatrix{ v\ \bullet \ar@/^.75pc/@{<-}[rr]^a \ar@/_.75pc/[rr]_a & & \bullet\ v }</m></span><br/>Try and consider this as a solid disk called {{M|P}} orientated clockwise<br/>(so the boundary of {{M|P}} is {{M|a+a}})</center> | ||
|- | |- | ||
− | ! Set up | + | ! Set up {{M|A}}, denoted {{M|\mathbb{RP}^2_A}} |
|} | |} | ||
− | </div>The chain complexes are <span style="font-size:1. | + | </div>The chain complexes are: <span style="font-size:1.25em;"><m>\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 }</m></span><br/> |
with: | with: | ||
# {{M|1=\partial_2(P)=a+a=2a}} | # {{M|1=\partial_2(P)=a+a=2a}} | ||
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# {{M|1=H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} } }} | # {{M|1=H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} } }} | ||
# {{M|1=H_2\cong 0}} | # {{M|1=H_2\cong 0}} | ||
+ | |||
+ | =={{M|\mathbb{RP}^2_B}}== | ||
+ | <div style="float:right;margin:0px;margin-right:0.2em;max-width:25em;"> | ||
+ | {| class="wikitable" border="1" style="margin:0px;" | ||
+ | | <center><span style="font-size:1.5em;"><m>\xymatrix{ w\ \bullet \ar@{<-}[rr]^a \ar@<.8ex>[d]_b & & \bullet\ v \ar@<-.8ex>@{<-}[d]^b \\ v\ \bullet \ar[rr]_a & & \bullet\ w}</m></span></center><br/>Text | ||
+ | |- | ||
+ | ! Set up {{M|B}}, denoted {{M|\mathbb{RP}^2_B}} | ||
+ | |} | ||
+ | </div>The chain complexes are: <span style="font-size:1.25em;"><m>\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 }</m></span><br/> | ||
+ | With: | ||
+ | # {{M|1=\partial_2(P)=2(a+b)}} | ||
+ | # {{M|\partial_1}} | ||
+ | #* {{M|1=\partial_1(a)=w-v}} | ||
+ | #* {{M|1=\partial_1(b)=v-w}} | ||
+ | # {{M|\partial_0}} | ||
+ | #* {{M|1=\partial_0(v)=0}} | ||
+ | #* {{M|1=\partial_0(w)=0}} | ||
+ | |||
+ | |||
+ | On paper I ended up with: | ||
+ | # {{M|1=H_2\cong 0}} | ||
+ | # {{M|1=H_1\cong\frac{\mathbb{Z} }{2\mathbb{Z} } }} | ||
+ | # {{M|1=H_0\cong\mathbb{Z} }} |
Revision as of 01:03, 16 October 2016
[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:
- [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]
Text |
Set up [ilmath]B[/ilmath], denoted [ilmath]\mathbb{RP}^2_B[/ilmath] |
---|
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]