This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime?
I understand the formal explanation given as an answer to Earthliŋ, namely, that the requirement needed for the action defining the spaces to be free. However, I would like to somehow see geometrically/visually why, if we take two numbers which are not coprime, the resulting space turns out not to be a manifold.
Any suggestions for intuition will be greatly appreciated.
