Editing Homotopy (section)

===Groups===

{{main|Homotopy group}}
Since the relation of two functions <math>f, g\colon X\to Y</math> being homotopic relative to a subspace is an equivalence relation, we can look at the [[equivalence class]]es of maps between a fixed ''X'' and ''Y''. If we fix <math>X = [0,1]^n</math>, the unit interval [0,&thinsp;1] [[cartesian product|crossed]] with itself ''n'' times, and we take its [[Boundary (topology)|boundary]] <math>\partial([0,1]^n)</math> as a subspace, then the equivalence classes form a group, denoted <math>\pi_n(Y,y_0)</math>, where <math>y_0</math> is in the image of the subspace <math>\partial([0,1]^n)</math>.

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