The Fundamental Group and Loops on a Circle
Loops based at a point, counted up to continuous deformation, form a group. Computing it for the circle is what proves a disc and an annulus are genuinely different spaces rather than merely drawn differently.
What a learner can do afterwards
- Decide whether two given loops are homotopic
- Explain why the fundamental group of the circle is the integers
- Use the group to prove that two spaces are not homeomorphic
1 · Read
A loop is a path that returns to its start, and the unit circle parametrizes position as cos t with sin t. Winding counts laps with direction as sign. The loop g of t = (cos 6 pi t, sin 6 pi t) for t in 0 to 1 sweeps angle 0 to 6 pi, and each 2 pi is one turn, so it winds 3 times.
Two loops are homotopic when one deforms into the other without breaking. On the circle, equal winding numbers mean homotopic: two loops winding twice each are homotopic to each other. A loop winding twice can never deform to a constant loop, since 2 differs from 0 and winding is invariant.
Loop classes form a group under concatenation, and for the circle that group is the integers. Running one loop after another adds the winding numbers, so winding 2 followed by winding 3 gives winding 5. Reversing a loop negates its number, so the reverse of winding 2 is winding -2.
The group tells genuinely different spaces apart. A disc has only contractible loops, while an annulus owns a loop around its hole that never shrinks. That loop witnesses a winding number no disc loop can carry, so the two spaces are not the same.
Winding classifies circle loops, concatenation adds the counts, and the resulting integers separate the disc from the annulus.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.