Difference between revisions of "Loop concatenation"
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| − | {{Stub page|grade=A*|msg=Stub page}} | + | {{Stub page|grade=A*|msg=Stub page. Be sure to link to [[concatenation of loops and paths (homotopy)]] and [[path concatenation (topology)]]}} |
__TOC__ | __TOC__ | ||
==Definition== | ==Definition== | ||
| − | Loop [[concatenation]] is a special case of | + | Loop [[concatenation]] is a special case of {{link|path concatenation|topology}}. We use {{M|1=I:=[0,1]\subset\mathbb{R} }} to denote the [[closed unit interval]] in [[reals|{{M|\mathbb{R} }}]]. |
Let {{Top.|X|J}} be a [[topological space]] and let {{M|\ell_1:I\rightarrow X}} and {{M|\ell_2:I\rightarrow X}} be [[loops]] in {{Top.|X|J}} based at {{M|b\in X}}<ref group="Note">That is: {{XXX|Put definition of loop based at {{M|b\in X}} here}}</ref>; then we can ''[[concatenate]]'' the loops: | Let {{Top.|X|J}} be a [[topological space]] and let {{M|\ell_1:I\rightarrow X}} and {{M|\ell_2:I\rightarrow X}} be [[loops]] in {{Top.|X|J}} based at {{M|b\in X}}<ref group="Note">That is: {{XXX|Put definition of loop based at {{M|b\in X}} here}}</ref>; then we can ''[[concatenate]]'' the loops: | ||
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See: [[The fundamental group]] for more information. | See: [[The fundamental group]] for more information. | ||
| + | ==See also== | ||
| + | * {{link|Concatenation of loops and paths|homotopy}} | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 09:17, 6 November 2016
Contents
[hide]Definition
Loop concatenation is a special case of path concatenation. We use I:=[0,1]⊂R to denote the closed unit interval in R.
Let (X,J) be a topological space and let ℓ1:I→X and ℓ2:I→X be loops in (X,J) based at b∈X[Note 1]; then we can concatenate the loops:
- ℓ1∗ℓ2:I→X by (ℓ1∗ℓ2):t↦{ℓ1(2t)for t∈[0,12]ℓ2(2t−1)for t∈[12,1][Note 2] - we claim this is also a loop based at b (see Claim 1)
- In words: the loop ℓ1∗ℓ2 first does ℓ1 but at double the speed, thus completing ℓ1 by t=12. Then, as ℓ1 ends at b we're in a position to start ℓ2. We do this at double speed, thus completing ℓ2 by time t=12.
Loops also lend themselves to other concatenations, all permutations of concatenations of ℓ1, ℓ−11, ℓ2 and ℓ−12 exist.
Caveats
Loop concatenation is not associative, that is:
- (ℓ1∗ℓ2)∗ℓ3≠ℓ1∗(ℓ2∗ℓ3)
Notice the loop (ℓ1∗ℓ2)∗ℓ3 does ℓ1 at 4x the normal speed, completing it by t=14, then embarks on ℓ2 at 4x the speed also, completing that by t=24=12, then embarks on ℓ3 at double speed, completing it by t=1.
Whereas, ℓ1∗(ℓ2∗ℓ3) does ℓ1 at double speed, completing it by t=12, then embarks on ℓ2 at 4x speed, completing it by t=34, then embarks on ℓ3 at 4x speed, completing it by t=1.
Although the image of both loops is the same (that is: ((ℓ1∗ℓ2)∗ℓ3)(I)=(ℓ1∗(ℓ2∗ℓ3))(I), they are clearly different. However (ℓ1∗ℓ2)∗ℓ3 and ℓ1∗(ℓ2∗ℓ3) are path homotopic, or homotopic rel {0,1}
See: The fundamental group for more information.
See also
Notes
- Jump up ↑ That is: TODO: Put definition of loop based at b∈X here
- Jump up ↑ We include t=12 in both parts as a nod to the pasting lemma.
