Definition:Loop (Topology)/Circle Representative of Loop

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\gamma: \closedint 0 1 \to S$ be a loop in $T$.

Let $\Bbb S^1 \subseteq \C$ be the unit circle in $\C$:

$\Bbb S^1 = \set {z \in \C : \size z = 1}$

Suppose $\omega : \closedint 0 1 \to \Bbb S^1$ such that $\map \omega s = \map \exp {2 \pi i s}$.

Then the unique map $\tilde f : \Bbb S^1 \to T$ such that $\tilde f \circ \omega = f$ is called the circle representative of $f$.

Sources

• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous): $\S 7$: Homotopy and the Fundamental Group. Circle Representatives