Regular and non-regular covering spaces of S1∨S1∨S1 \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} .

I tried to draw the regular and non-regular covering spaces of S1S1S1. I think the regular covering space is:

Covering spaces

Is it true? How do you draw the non-regular covering space of this one?

Answer

The example you drew is not a covering space of S1S1S1, because the unique vertex of S1S1S1 has valence 6, and therefore each vertex of the covering space must also have valence 6, but the vertices of your graph have valence 4.

Furthermore, the six directional rays of P can be labelled a, a1, b, b1, c, c1, and so each of the six directional rays at each vertex of the covering space must also have those labels.

Using this idea you can draw many covering spaces, and if you try it out you will discover many regular ones and many nonregular ones. Here is the method.

Choose an integer D for the degree of the covering space. Draw D points p1,,pD which will cover the base point P. For each of the points pi, draw six directional rays at pi labelled a, a1, b, b1, c, c1, which will cover the directional rays at P. Now you have a collection of D six-pointed stars, and amongst them are D directional rays labelled a, and D directional rays labelled a1, et cetera.

Now choose any bijection between the set of D directional rays labelled a and the set of D direction rays labelled a1 directional rays. Connect each pair of rays with an edge labelled a (if you are doing this on paper you will need to allow for edges to cross over and under each other). Similarly, choose a bijection between the b and b1 directional rays and connect each pair with a b edge, and similarly for c and c1. The result is a covering space of S1S1S1.

Attribution
Source : Link , Question Author : Mücahit Meral , Answer Author : Lee Mosher

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