Let (X,x0) be a pointed topological space (that is, a topological space with a chosen basepoint x0). Denote by [(S1,1),(X,x0)]the set of homotopy classes of maps σ:S1→Xsuch that σ(1)=x0. Here, 1 denotes the basepoint (1,0)∈S1. Define a product [(S1,1),(X,x0)]×[(S1,1),(X,x0)]→[(S1,1),(X,x0)]by [σ][τ]=[στ], where στ means “travel along σ and then τ”. This gives [(S1,1),(X,x0)] a group structure and we define the fundamental group of (X,x0)to be π1(X,x0)=[(S1,1),(X,x0)].
In general, the fundamental group of a topological space depends upon the choice of basepoint. However, basepoints in the same path-component of the space will give isomorphic groups. In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphism, without the need to specify a basepoint.
Here are some examples of fundamental groups of familiar spaces:
• π1(ℝn)≅{0} for each n∈ℕ.
• π1(S1)≅ℤ.
• π1(T)≅ℤ⊕ℤ, where T is the torus.
It can be shown that π1 is a functorfrom the category of pointed topological spaces to the category of groups. In particular, the fundamental group is a topological invariant, in the sense that if (X,x0) is homeomorphic to (Y,y0) via a basepoint-preserving map, then π1(X,x0) is isomorphic to π1(Y,y0).
It can also be shown that two homotopically equivalent path-connected spaces have isomorphic fundamental groups.
