Difference between revisions of "Homotopy invariance of path concatenation"

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(Created page with "{{Stub page|grade=A*|msg=Really not in the mood for this, done it anyway, check first and flesh out}} __TOC__ ==Statement== File:HomotopyInvarianceOfPathConcatenation.JPG|th...")
 
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==Statement==
 
==Statement==
 
[[File:HomotopyInvarianceOfPathConcatenation.JPG|thumb|{{XXX|This caption}}]]
 
[[File:HomotopyInvarianceOfPathConcatenation.JPG|thumb|{{XXX|This caption}}]]
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Let {{M|p_1,p_2,p_1',p_2'\in}}{{C(I,X)}} be given. Suppose {{M|H:\ p_1\simeq p_1'\ (\text{rel }\{0,1\})}} and {{M|H:\ p_2\simeq p_2'\ (\text{rel }\{0,1\})}} are {{plural|end point preserving homotop|y|ies}} (where {{M|H_1,H_2:[0,1]\times [0,1]\rightarrow X}} are the specific {{plural|homotop|y|ies}} of the {{link|path|topology|s}}) then{{rITTMJML}}:
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Let {{M|p_1,p_2,p_1',p_2'\in}}{{C(I,X)}} be given. Suppose {{M|H_1:\ p_1\simeq p_1'\ (\text{rel }\{0,1\})}} and {{M|H_2:\ p_2\simeq p_2'\ (\text{rel }\{0,1\})}} are {{plural|end point preserving homotop|y|ies}} (where {{M|H_1,H_2:[0,1]\times [0,1]\rightarrow X}} are the specific {{plural|homotop|y|ies}} of the {{link|path|topology|s}}) then{{rITTMJML}}:
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* {{M|H:p_1*p_2\simeq p_1'*p_2'\ (\text{rel }\{0,1\})}} where {{M|1=H:=H_1*H_2}} - the [[homotopy concatenation]], explicitly:
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* {{M|H:p_1*p_2\simeq p_1'*p_2'\ (\text{rel }\{0,1\})}} where
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** {{M|1=H:[0,1]\times[0,1]\rightarrow X}} by {{M|1=H:(s,t)\mapsto\left\{H1(s,2t)for t∈[0,12]H2(s,2t−1)for t∈[12,1]\right.}}  
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** {{M|p_1*p_2}} denotes {{link|path concatenation|topology}}, explicitly:
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*** Note that the fact {{M|1=t=\frac{1}{2} }} is in both parts is a nod towards the use of the [[pasting lemma]]
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*** {{M|1=p_1*p_2:[0,1]\rightarrow X}} by {{M|p_1*p_2:t\mapsto\left\{p1(2t)for t∈[0,12]p2(2t−1)for t∈[12,1]\right. }}
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**** Note that the fact {{M|1=t=\frac{1}{2} }} is in both parts is a nod towards the use of the [[pasting lemma]]
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** {{M|1=H:=H_1*H_2}} - the [[homotopy concatenation]], explicitly:
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*** {{M|1=H:[0,1]\times[0,1]\rightarrow X}} by {{M|1=H:(s,t)\mapsto\left\{H1(s,2t)for t∈[0,12]H2(s,2t−1)for t∈[12,1]\right.}}  
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**** Note that the fact {{M|1=t=\frac{1}{2} }} is in both parts is a nod towards the use of the [[pasting lemma]]
 
<div style="clear:both;"><div>
 
<div style="clear:both;"><div>
 
==Proof==
 
==Proof==

Latest revision as of 19:11, 9 November 2016

Stub grade: A*
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Really not in the mood for this, done it anyway, check first and flesh out

Statement

TODO: This caption

Let p1,p2,p′1,p′2∈C([0,1],X) be given. Suppose H1: p1≃p′1 (rel {0,1}) and H2: p2≃p′2 (rel {0,1}) are end point preserving homotopies (where H1,H2:[0,1]×[0,1]→X are the specific homotopies of the paths) then[1]:

  • H:p1∗p2≃p′1∗p′2 (rel {0,1}) where
    • p1∗p2 denotes path concatenation, explicitly:
      • p1∗p2:[0,1]→X by p1∗p2:t↦{p1(2t)for t∈[0,12]p2(2t−1)for t∈[12,1]
        • Note that the fact t=12 is in both parts is a nod towards the use of the pasting lemma
    • H:=H1∗H2 - the homotopy concatenation, explicitly:
      • H:[0,1]×[0,1]→X by H:(s,t)↦{H1(s,2t)for t∈[0,12]H2(s,2t−1)for t∈[12,1]
        • Note that the fact t=12 is in both parts is a nod towards the use of the pasting lemma

Proof

Grade: A*
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It's basically already done. All we have to show is that the homotopy concatenation, H, fits the requirements

See also

References

  1. Jump up ↑ Introduction to Topological Manifolds - John M. Lee
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