I need to calculate the fundamental group of the space obtained by punching k holes in the sphere.
I understand the spaces H(p) and M(q), with H(p) being the sphere with p handles added and M(q) as the sphere with q Mobius strips sewn in, and I believe I understand their fundamental groups and sort of how to calculate them. I just can't seem to understand how these concepts relate if they do, and therefore, how to find the fundamental group of the space obtained by punching k holes in the sphere.