# 3-fold covers of the circle

• Feb 16th 2010, 01:00 PM
cribby
3-fold covers of the circle
I'm trying to understand an example in notes from a day I missed in class. There is a collection of diagrams that are supposed to be the 3-fold covers of the circle, S1.

The first collection is a set of 3 disjoint circles, each with a marked base-point. I presume this is to mean that one 3-fold cover of S1 is the disconnected union of three copies of S1 (something like E = S1 X {1, 2, 3}), and the marked base-points just represents the inverse image of the covering map of an arbitrary point of S1.

The second collection consists of two figures: a copy of S1 (w/ base-point) and a crescent (w/ 2 base points). I am less certain here, but I think the first figure is, of course, a copy of S1 and the second figure is a 2-fold cover of S1 (something like two "cycles" on the helix but with joined endpoints?). The 3-fold cover of S1 here then would be their disconnected union.

The third is a single figure, a triangle with the vertices marked. I presume this is a connected 3-fold cover of S1 (something like three "cycles" on the helix, with joined endpoints?).

Do I understand the gist of this example?
• Feb 16th 2010, 02:50 PM
cribby
Ok, I think I get it now.

I would still like to know how these covering spaces are denoted non-diagramatically. I must've missed two pretty informative lectures to be so lost right now. Basically I feel that I am lacking any formal, complete examples of covering spaces (specifically constructing them). The last things in my notes regarding covering spaces were, well, the definition.