The Klein bottle and the torus both have $\mathbf{R}^2$ as universal cover, and the torus can cover the Klein bottle.

Does this always happen? If $A$ and $B$ have the same universal cover, must $A$ cover $B$ or vice versa?

EDIT: as pointed out below the answer is no. So my new question is: is there any hypotheses that can be added to make it true?

  • $\begingroup$ The figure-eight and the space consisting of three circles touching each other is a counterexample, unless I misrecall the definition of cover space... $\endgroup$
    – Arthur
    Sep 24 '13 at 19:13
  • 3
    $\begingroup$ @Arthur In fact, the 3 circles cover the figure-8 $\endgroup$
    – user8268
    Sep 24 '13 at 19:16

No. For example, let $A$ be the union of 3 equally big circles touching each other, with the centers on the same line, and $B$ the same, but with $4$ circles. Both spaces are graphs with 4-valent vertices, $A$ has 4 edges and $B$ 6, so none of them can cover the other one. Yet they have the same universal covering space (4-valent tree, the universal covering space of $\infty$).


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