Geometric intuition of covering spaces

Question

Can somebody give me a geometrical intuition of the pullback of a covering space? For example: Can the $2$-fold covering of the Klein bottle by the torus be pulled back to a covering of the mobius strip by a cylinder? Looking at the planar representations of these spaces I think this might be possible but I really get messed up with this...

Answer 1

Sure. The mobius strip embeds nicely in the Klein bottle; the "pullback" in this case is just the preimage of that strip under the projection map, which turns out to be a cylinder, which is (I think) a thin-band neighborhood of a curve that wraps twice around one "direction" on the torus and once around the other, i.e., a "tubular neighborhood" of a (2, 1) torus knot.