Difference between revisions of "Notes:Homology/Real projective plane"

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(Fixed some typos, added "skeleton" of second "set up" for RP2)
m (Weird. Overflow now fixed. Guess I can't use xymatrix in a span tag!)
 
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=={{M|\mathbb{RP}^2_A}}==
 
=={{M|\mathbb{RP}^2_A}}==
 
<div style="float:right;margin:0px;margin-left:0.2px;overflow:hidden;">
 
<div style="float:right;margin:0px;margin-left:0.2px;overflow:hidden;">
−
{| class="wikitable" border="1" style="margin:0px;max-width:25em;"
+
{| class="wikitable" border="1" style="margin:0px;max-width:25em;overflow:hidden;"
 
|-
 
|-
−
| <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;overflow:hidden;"><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 {{M|A}}, denoted {{M|\mathbb{RP}^2_A}}
 
! Set up {{M|A}}, denoted {{M|\mathbb{RP}^2_A}}
 
|}
 
|}
−
</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/>
+
</div>The chain complexes are: <div style="font-size:1.25em;overflow:hidden;"><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></div>
 
with:
 
with:
 
# {{M|1=\partial_2(P)=a+a=2a}}
 
# {{M|1=\partial_2(P)=a+a=2a}}
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=={{M|\mathbb{RP}^2_B}}==
 
=={{M|\mathbb{RP}^2_B}}==
−
<div style="float:right;margin:0px;margin-right:0.2em;max-width:25em;">
+
<div style="float:right;margin:0px;margin-right:0.2em;max-width:25em;overflow:hidden;">
−
{| class="wikitable" border="1" style="margin:0px;"
+
{| class="wikitable" border="1" style="margin:0px;overflow:hidden;"
 
| <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
 
| <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}}
 
! 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/>
+
</div>The chain complexes are: <div style="font-size:1.25em;overflow:hidden;"><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></div><br/>
 
With:
 
With:
 
# {{M|1=\partial_2(P)=2(a+b)}}
 
# {{M|1=\partial_2(P)=2(a+b)}}

Latest revision as of 01:48, 16 October 2016

[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]


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]
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