How i see this the homotopy group of order 1 is giving information about the "holes" in a topological space. In that way what kind of information is giving the homotopy group of order 2 or n in general?
I can't think pretty clear about it, but in the first case it's seems clear that a hole is defining different equivalent classes of homotopic loops in the space.
But the FG of order 2 is like homotopic classes of loops in the loop space of the space X, so is basically holes in the loop space, and this actually is like: "The actual points for the loop are in the set but you can't walk through the loop in some cases", a loop passing in some points can be absent but other loop passing thru the same points with different parametrization can be there, so, is not about "holes" in the real space, but like "phantom holes" that are just holes in the loop space. Is hard to imagine this.
So, how do you imaging the info giving by the homotopy group of order 2 of X (or n in general) in the space X?