Robert's Mistakes

diagram 7: covering map / covering graph

Aug 18, 2013

Last time we saw a way to "simplify" a Klein bottle by unfolding it into a square. The figure eight graph B below can be interpreted as an even simpler picture, this time for a two-holed torus. It says that, given a starting point a, there are two loops wich cannot be shrunk to a point.

Just above there is a different but, in a special way, similar graph X. It has two vertices and four edges. Between those graphs we have an arrow p representing a projection from the upper to the lower graph. Here, two pairs of edges in X are mapped to the ones with the same label in B, the vertices x and x' both go to a.

Funny thing is, that arrow points from the complicated X to the simple B, yet we speak of B as generating X. This is because given some X and B, projections are usually easy to describe (and to think of). Another funny thing is that the fundamental group of X is considered a subgroup of B, again just the other way around than one might have expected. Here's why:

The fundamental group describes all possible (nontrivial or "unshrinkable") paths we can have on some topological surface. So winding twice around a handle is different from winding around once. The fundamental group of B contains elements like aaa, (-b)ab(-a)bb, or any other word consisting of a, b and their negatives.

To study topological spaces, we usually pick a starting point (mostly for conveniece). Let's pick x to study X. The resulting fundamental group is built of aa, bb, and ab (because we have to get back after any single letter) and that's less than the set of all words, as allowed for the space B.

The picture is part of a screenshot from N.J. Wildbergers AlgTop series, this time part 29 around 12:50. I promise, i will get you more diverse links in the future. See, not all is topology and i'm not an addict, i can stop doing topology anytime i want to... AlgTop29: Universal covering spaces

For prettier images, more examples and a continuous version: