Homotopy Group - Introduction

Introduction

In modern mathematics it is common to study a category by associating to every object of this category a simpler object which still retains a sufficient amount of information about the object in question. Homotopy groups are such way of associating groups to topological spaces.

That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they can't have the same topological structure—a fact which may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.

As for the example: the first homotopy group of the torus T is

π₁(T)=Z2,

because the universal cover of the torus is the complex plane C, mapping to the torus T ≅ C / Z2. Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand the sphere S2 satisfies

π₁(S2)=0,

because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more complicated examples of homotopy groups).

Hence the torus is not homeomorphic to the sphere.