Homotopy - Homotopy Groups

Homotopy Groups

Since the relation of two functions f, g : X → Y being homotopic relative to a subspace is an equivalence relation, we can look at the equivalence classes of maps between a fixed X and Y. If we fix X = n, the unit interval crossed with itself n times, and we take a subspace to be its boundary ∂(n) then the equivalence classes form a group, denoted π_n(Y,y₀), where y₀ is in the image of the subspace ∂(n).

We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy groups. In the case n = 1, it is also called the fundamental group.