What is the univesal cover of 2-sphere for which the two poles are connected to each other with a line? How should i compute the universal cover of 2-sphere for which the two poles are connected to each other with a line?
I think it should be $ S^2 $ but it looks like a ruined doughnut and the universal cover for doughnut is $ R^2$

You are correct about the space looking like a ruined donut.
The universal covering space of this is infinitely many spheres (in both directions), such that the North pole of the i$^{\text{th}}$ sphere is glued to the South pole of the (i+1)$^{\text{st}}$ sphere.
The covering map just maps each sphere down.
Edit: As Roy says, this space is only homotopy equivalent to the universal cover. Inserting a line segment between each sphere gives the correct answer.
