Loop concatenation

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Stub page. Be sure to link to concatenation of loops and paths (homotopy) and path concatenation (topology)

Definition

Loop concatenation is a special case of path concatenation. We use $I:=[0,1]\subset\mathbb{R}$ to denote the closed unit interval in $\mathbb{R}$.

Let $(X,\mathcal{ J })$ be a topological space and let $\ell_1:I\rightarrow X$ and $\ell_2:I\rightarrow X$ be loops in $(X,\mathcal{ J })$ based at $b\in X$; then we can concatenate the loops:

• $\ell_1*\ell_2:I\rightarrow X$ by $(\ell_1*\ell_2):t\mapsto\left\{\begin{array}{lr}\ell_1(2t) & \text{for }t\in[0,\frac{1}{2}] \\ \ell_2(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.$ - we claim this is also a loop based at $b$ (see Claim 1)
  • In words: the loop $\ell_1*\ell_2$ first does $\ell_1$ but at double the speed, thus completing $\ell_1$ by $t=\frac{1}{2}$. Then, as $\ell_1$ ends at $b$ we're in a position to start $\ell_2$. We do this at double speed, thus completing $\ell_2$ by time $t=\frac{1}{2}$.

Loops also lend themselves to other concatenations, all permutations of concatenations of $\ell_1$, $\ell_1^{-1}$, $\ell_2$ and $\ell_2^{-1}$ exist.

Caveats

Loop concatenation is not associative, that is:

• $(\ell_1*\ell_2)*\ell_3\ne\ell_1*(\ell_2*\ell_3)$

Notice the loop $(\ell_1*\ell_2)*\ell_3$ does $\ell_1$ at 4x the normal speed, completing it by $t=\frac{1}{4}$, then embarks on $\ell_2$ at 4x the speed also, completing that by $t=\frac{2}{4}=\frac{1}{2}$, then embarks on $\ell_3$ at double speed, completing it by $t=1$.

Whereas, $\ell_1*(\ell_2*\ell_3)$ does $\ell_1$ at double speed, completing it by $t=\frac{1}{2}$, then embarks on $\ell_2$ at 4x speed, completing it by $t=\frac{3}{4}$, then embarks on $\ell_3$ at 4x speed, completing it by $t=1$.

Although the image of both loops is the same (that is: $\big((\ell_1*\ell_2)*\ell_3\big)(I)=\big(\ell_1*(\ell_2*\ell_3)\big)(I)$, they are clearly different. However $(\ell_1*\ell_2)*\ell_3$ and $\ell_1*(\ell_2*\ell_3)$ are path homotopic, or homotopic $\text{rel }\{0,1\}$

See: The fundamental group for more information.

See also

• Concatenation of loops and paths