# Universal cover of a figure eight?

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'.

Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can be visualized by the Cayley graph'' but I can't see how this graph is simply connected?

NOR, how I am supposed to draw it?

Is there a simple 'draw-able' universal cover of the figure-eight?

• Attempting to draw it completely could leave you "cross." Commented Apr 7, 2013 at 17:48
• I'll give you an idea in a comment. If you want it fleshed out, I'll turn it into an answer. The universal cover of each circle, independently, is a helix. How to "glue" together these helixes to accommodate our case? Start with one helix, loops on the left circle. At regularly spaced points on this helix, attach another helix heading in another direction, these are points where you start moving on the other circle. On each of these loops, attach another (smaller) helix at regularly spaced intervals, which are points where you move back to swirling around the first circle. Repeat and shrink. Commented Apr 7, 2013 at 17:53
• Note that you always have to add more and more smaller helixes. This is because the fundamental group is the free group on two generators. Once you have entered a smaller helix from a bigger helix you can only return backwards by the direct inverse path. Commented Apr 7, 2013 at 17:54
• You can find the example in Hatcher's book. Commented Apr 7, 2013 at 18:19

## 2 Answers

Here is a picture of the universal cover of a figure eight I made for a topology course in 1991 (!), using Macintosh-Pascal. Note the self-similarity of the "ends".

• Does this idea generalize to infinite growing circles joined at a single point? (I am referring to an infinite wedge sum, not to Hawaiian Earrings) Commented Dec 7, 2019 at 6:14
• When you mean infinitely many loops sharing a single common point $P$ you obtain a similar figure, but instead of four you have infinitely many edges ending at each copy of $P$. Commented Dec 7, 2019 at 10:19

The universal covering space is unique up to homeomorphism. Remember the fundamental group of the figure 8 is the non-commutative free group on two elements. Hence the complex appearance of its covering space.