How do you compute the Fundamental group of the 1-skeleton of the 3-cube = ? What about the Fundamental group of the 1- skeleton of the 4-cube ?
I know the Fundamental group of a space X at a point is the set of homotopy classes of loops of X based at . 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?
1-skeletons of cubes are graphs, and to compute the fundamental group of a graph (which is always free) you just take a maximal tree, homotope it down to a point, and what you're left with is a wedge product of n circles. So your fundamental group is free on n generators.
For the 1-skeleton of the 3-cube, I can visualize by drawing that there will be 5 edges not included in the maximal tree of the graph, so the fundamental group of the 1-skeleton of the 3-cube is the free product of 5 copies of Z .
But what about for the 1-skeleton of the 4-cube ? Since I cannot visualize it, I am not sure how many edges won't be in the maximal tree(number of circuits) , so I am not sure how the fundamental group is going to look like, same thing with the n-cube .