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Let $\mathbb{T}$ be the 2-dimensional torus and $p,q$ two distinct points of $\mathbb{T}$.

The problem is to find a non-regular 3-covering of the torus having two of its points identified, namely

$$\mathbb T /_{p\text{~}q}$$.

A regular 3-cover is relatively easy to find

Here is a regular 3-covering where the 3 tori touch each other at one point

However I am also wondering if the following is a 3-covering space which is not normal The first and the thrird torus are deformed a bit so that they touch at the point B. They touch with the second torus at the points A and C respectively

Naturally, one would want to gain inspiration from a non-regular 3-covers of, say, the bouquet of 2 circles, however I am unable to do so.

P.S. Sorry for the bad drawings.

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maybe, this one? glued points are connected by the lines. enter image description here every automorphism of the covering will map the parallel of the long torus to itself (preserving orientation), also it preserves each of pairs of glued points, so it must be identity.

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  • $\begingroup$ Thank you for your answer Andrey. I will give you the bounty, but could you please check if my solution is correct (since it was my initial question). Thanks. $\endgroup$ Commented Jun 6, 2017 at 0:26

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