Liftings of Nullhomotopic Maps

Question

Is the lifting of a nullhomotopic map always a loop in a path connected space? This question is perhaps very trivial,but I am struggling with it. Please help.

Answer 1

Yes, assuming you meant to ask about maps of the circle which are null homotopic, and you can find a proof in any textbook covering covering theory, say, Massey or Hatcher.

Answer 2

No. If you lift the map from $I$ to $S^1$, that goes around the circle once, to the covering of $S^1$ by $\mathbb{R}$, then the image will be the straight line path from $0$ to $1$.