Algebraic Topology: Fundamental group of a cube

[math8]

How do you compute the Fundamental group of the 1-skeleton of the 3-cube [tex]I^{3}[/tex] = [tex][0,1]^{3}[/tex] ? What about the Fundamental group of the 1- skeleton of the 4-cube [tex]I^{4}[/tex] ?

I know the Fundamental group of a space X at a point [tex]x_{0}[/tex] is the set of homotopy classes of loops of X based at [tex]x_{0}[/tex] . And that the 1-skeleton of a space X is the union of all cells of the CW complex for X up to dimension 1. But how do you find the fundamental group of the 1 skeleton for those cubes?

 

[lippyka]

For the 3-cube I^{3}, the 1-skeleton consists of all points and edges of the cube. The fundamental group of this space can be computed as follows:Let x_{0} be a point in the 1-skeleton. Then the fundamental group of the 1-skeleton of the 3-cube at x_{0} is the free group generated by the 8 edges that form the cube. Thus, the fundamental group of the 1-skeleton of the 3-cube is F_{8}.For the 4-cube I^{4}, the 1-skeleton consists of all points and edges of the cube. The fundamental group of this space can be computed as follows:Let x_{0} be a point in the 1-skeleton. Then the fundamental group of the 1-skeleton of the 4-cube at x_{0} is the free group generated by the 24 edges that form the cube. Thus, the fundamental group of the 1-skeleton of the 4-cube is F_{24}.