fundamental group

Let (X,x0) be a pointed topological spacePlanetmathPlanetmath (that is, a topological spaceMathworldPlanetmath with a chosen basepoint x0). Denote by [(S1,1),(X,x0)] the set of homotopy classes of maps σ:S1X such that σ(1)=x0. Here, 1 denotes the basepoint (1,0)S1. Define a productMathworldPlanetmathPlanetmath [(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 groupMathworldPlanetmathPlanetmath 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 groupsMathworldPlanetmath. In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, without the need to specify a basepoint.

Here are some examples of fundamental groups of familiar spaces:

  • π1(n){0} for each n.

  • π1(S1).

  • π1(T), where T is the torus.

It can be shown that π1 is a functorMathworldPlanetmath from 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 homeomorphicMathworldPlanetmath 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.

Homotopy groupsMathworldPlanetmath generalize the concept of the fundamental group to higher dimensionsMathworldPlanetmath. The fundamental group is the first homotopy group, which is why the notation π1 is used.

Title fundamental group
Canonical name FundamentalGroup
Date of creation 2013-03-22 11:58:44
Last modified on 2013-03-22 11:58:44
Owner yark (2760)
Last modified by yark (2760)
Numerical id 16
Author yark (2760)
Entry type Definition
Classification msc 57M05
Classification msc 55Q05
Classification msc 20F34
Synonym first homotopy group
Related topic Group
Related topic Curve
Related topic EtaleFundamentalGroup