---
title: "The Fundamental Group and Loops on a Circle"
description: "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 dif"
canonical: https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4
source: https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Topology · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4

## Ready when they can

- 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

## Lesson: Counting laps around a circle

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Winding classifies circle loops, concatenation adds the counts, and the resulting integers separate the disc from the annulus.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Galois Correspondence](https://lightmysky.com/learn/mathematics/the-galois-correspondence-mt_30kyliQR_L)
- [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L)
- [Compactness and Connectedness](https://lightmysky.com/learn/mathematics/compactness-and-connectedness-mt_ybV1lUedHs)