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Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]b\in X[/ilmath] be given. Then [ilmath]\Omega(X,b)\subseteq[/ilmath][ilmath]C([0,1],X)[/ilmath] is the set containing all loops based at [ilmath]b[/ilmath][1]. That is:

  • [ilmath](\ell:I\rightarrow X)\in\Omega(X,b)[/ilmath] means [ilmath]\ell[/ilmath] is a loop based at [ilmath]b\in X[/ilmath] (AKA: a path such that [ilmath]\ell(0)=\ell(1)=b[/ilmath])

There is additional structure we can imbue on [ilmath]\Omega(X,b)[/ilmath]:

  • [ilmath]*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)[/ilmath] - the operation of loop concatenation:
    • [ilmath]*:(\ell_1,\ell_2)\mapsto\left((\ell_1*\ell_2):[0,1]\rightarrow X\text{ 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.\right)[/ilmath]

Caution:This is not a monoid or even a semigroup as [ilmath]*[/ilmath] is not associative. See "Caveats" below

This set and the operation of loop concatenation are a precursor for the fundamental group


Associativity (or lack of)

Note that for [ilmath]\alpha,\beta,\gamma\in\Omega(X,b)[/ilmath] that [ilmath]\alpha*(\beta*\gamma)\ne(\alpha*\beta)*\gamma[/ilmath], that is because [ilmath]\alpha*(\beta*\gamma)[/ilmath] spends [ilmath]0\le t\le \frac{1}{2} [/ilmath] doing [ilmath]\alpha[/ilmath] at double speed, then does [ilmath]\beta[/ilmath] during [ilmath]\frac{1}{2}\le t\le \frac{3}{4} [/ilmath] at 4x the normal speed, then lastly [ilmath]\gamma[/ilmath] during [ilmath]\frac{3}{4}\le t\le 1[/ilmath] at 4x the normal speed also.

In contrast, [ilmath](\alpha*\beta)*\gamma[/ilmath] does [ilmath]\alpha[/ilmath] at 4x normal speed during [ilmath]0\le t\le \frac{1}{4}[/ilmath] then [ilmath]\beta[/ilmath] at 4x normal speed during [ilmath]\frac{1}{4}\le t\le \frac{1}{2} [/ilmath] then lastly, [ilmath]\gamma[/ilmath] at double speed during [ilmath]\frac{1}{2}\le t\le 1[/ilmath]

These are clearly different functions

See also


  1. Introduction to Topological Manifolds - John M. Lee