Fundamental Group - Examples

Examples

Trivial fundamental group. In Euclidean space Rn, or any convex subset of Rn, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. A path-connected space with a trivial fundamental group is said to be simply connected.

Infinite cyclic fundamental group. The circle. Each homotopy class consists of all loops which wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop which winds around m times and another that winds around n times is a loop which winds around m + n times. So the fundamental group of the circle is isomorphic to, the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.

Free groups of higher rank: Graphs or punctured plane. Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian. For example, the fundamental group of the figure eight is the free group on two letters. More generally, the fundamental group of any graph is a free group. If the graph G is connected, then the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic of G.

The fundamental group of the plane punctured at n points is also the free group with n generators. The ith generator is the class of the loop that goes around the ith puncture without going around any other punctures.

Knot theory. A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot in R3.