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\{\begin{array}{lr}H_1(s,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(s,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\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\{\begin{array}{lr}p_1(2t)&\text{for }t\in[0,\frac{1}{2}]\\p_2(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\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\{\begin{array}{lr}H_1(s,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(s,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\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 [ilmath]p_1,p_2,p_1',p_2'\in[/ilmath][ilmath]C([0,1],X)[/ilmath] be given. Suppose [ilmath]H_1:\ p_1\simeq p_1'\ (\text{rel }\{0,1\})[/ilmath] and [ilmath]H_2:\ p_2\simeq p_2'\ (\text{rel }\{0,1\})[/ilmath] are end point preserving homotopies (where [ilmath]H_1,H_2:[0,1]\times [0,1]\rightarrow X[/ilmath] are the specific homotopies of the paths) then[1]:

  • [ilmath]H:p_1*p_2\simeq p_1'*p_2'\ (\text{rel }\{0,1\})[/ilmath] where
    • [ilmath]p_1*p_2[/ilmath] denotes path concatenation, explicitly:
      • [ilmath]p_1*p_2:[0,1]\rightarrow X[/ilmath] by [ilmath]p_1*p_2:t\mapsto\left\{\begin{array}{lr}p_1(2t)&\text{for }t\in[0,\frac{1}{2}]\\p_2(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right. [/ilmath]
        • Note that the fact [ilmath]t=\frac{1}{2}[/ilmath] is in both parts is a nod towards the use of the pasting lemma
    • [ilmath]H:=H_1*H_2[/ilmath] - the homotopy concatenation, explicitly:
      • [ilmath]H:[0,1]\times[0,1]\rightarrow X[/ilmath] by [ilmath]H:(s,t)\mapsto\left\{\begin{array}{lr}H_1(s,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(s,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.[/ilmath]
        • Note that the fact [ilmath]t=\frac{1}{2}[/ilmath] is in both parts is a nod towards the use of the pasting lemma

Proof

Grade: A*
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
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The message provided is:
It's basically already done. All we have to show is that the homotopy concatenation, [ilmath]H[/ilmath], fits the requirements

See also

References

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