The Fundamental Group and Loops on a Circle

You have used continuous maps to study compactness and connectedness. A path in a space X is a continuous map g from [0,1] to X. A loop based at b satisfies g(0)=g(1)=b. Our circle loops use b=(1,0) unless another base point is stated. The unit circle has points (cos theta, sin theta), with theta in radians.

A based homotopy from g to h is a continuous map H(t,s) from [0,1] x [0,1] into X. It satisfies H(t,0)=g(t), H(t,1)=h(t), and H(0,s)=H(1,s)=b for every s. The parameter t runs along a loop; s changes that loop. Both endpoints stay at b, and every intermediate loop stays in X.

Every circle path admits a continuous real angle theta(t) with g(t)=(cos theta(t), sin theta(t)). This angle lift records turns without resetting after a lap. For a loop, winding is (theta(1)-theta(0))/(2 pi), an integer. Counterclockwise is positive and clockwise is negative. Changing the starting lift by a multiple of 2 pi does not change this difference.

Based homotopy preserves winding for circle loops at a common base point. Conversely, interpolate angle lifts with matching starting angles. Equal windings make their ending angles agree too, keeping the base point fixed. This classifies the loops. A constant loop has winding 0; null-homotopic means homotopic to it.

The fundamental group's elements are based homotopy classes of loops. Concatenation runs one loop then another. For the circle, winding bijectively matches classes with integers and takes concatenation to addition. A bijection preserving the operation is an isomorphism. Every integer n occurs in (cos(2 pi n t), sin(2 pi n t)). The identity is the constant-loop class, of winding 0.

Concatenation adds windings: n+m. Reversal negates winding and gives the inverse class. Repeating a loop k times gives k times its winding. For a missing winding, subtract the known winding from the target: after winding -2, target 4 requires 4-(-2)=6. A path followed by its exact reverse has zero angle change, hence is null-homotopic on the circle.

A continuous change of timing can preserve a loop's class. Let rho map [0,1] continuously into [0,1], with rho(0)=0 and rho(1)=1. Then g(rho(t)) is based homotopic to g(t). The formula H(t,s)=g((1-s)t+s rho(t)) gives the homotopy: its input stays in [0,1], and both endpoints remain fixed. For example, rho(t)=t cubed changes timing without changing winding.

Take the closed unit disc D={x in R squared: ||x||<=1}, where ||x|| is distance from the origin. Every loop g in D contracts to its base point b using H(t,s)=(1-s)g(t)+s b. Straight segments stay in the convex disc, and both endpoints stay at b. Hence the disc's fundamental group is trivial, with one element. The same contraction works in a closed interval.

For the annulus A={x in R squared: 1<=||x||<=2}, use H(x,s)=((1-s)+s/||x||)x. A point of radius r moves radially with radius (1-s)r+s, which stays in [1,2] and ends at 1. Points on the inner circle stay fixed. This is a deformation retraction: a continuous deformation onto a subspace that fixes that subspace throughout. It induces an isomorphism of fundamental groups, so the annulus's group is the integers.

A homeomorphism induces an isomorphism of fundamental groups at corresponding base points. The disc's group has one element, while the annulus's group is isomorphic to the integers. These groups are not isomorphic, so the spaces are not homeomorphic. A loop going once around the annulus's hole represents winding 1, a nontrivial class that cannot contract inside the annulus.

The converse does not follow: isomorphic fundamental groups alone do not prove homeomorphism. A disc and [0,1] both have trivial groups. Yet deleting an interior point disconnects [0,1], while deleting any one point leaves the disc connected. A homeomorphism would restrict to a homeomorphism after deleting corresponding points, so these spaces are not homeomorphic.