This blog post will be about the following animation, that I had created a while ago. Through this blog post, I will try to explain what I am trying to explain in the given visual (due to the large size of the GIF, it may take some time for it to load completely on your web-browser). Here is the topic in hand:
There are several pieces of information that are being conveyed here. I will go through them individually. I will assume that the reader is familiar with basic topology.
What we are considering here is the Bethe Lattice of coordination number 4. A Bethe Lattice is a particular type of Cayley Graph, and the case that we are considering is, in fact, the most clearly given example on Wikipedia.
The Cayley graph of the free group on two generators a and b (picture from Wikipedia). Although the graph pictured below looks finite, the picture represents a tree with infinitely many nodes, built recursively from a chosen root.
For our purpose, we want to consider this infinite graph as a topological space. We will consider the topology for this Cayley graph as if it is a Simplicial 2-Complex, or to simply put, we will imagine this graph to be infinitely many line segments glued togeather at common vertices.
I will denote this topological space by .
Wedge of two circles
The wedge of two circles, or the rose with two petals, or the figure-8 space is the just the subset of given by any pair of circles sharing exactly one point. The topology on this set, of course, is the induced topology. Here is a picture below for the sake of clarity.
I will call this topological space .
Covering map, covering space
For a topological space , a covering space of is a topological space coupled with a continous map such that every has a neighbourhood around such that is a collection of disjoint open sets and each one of those open sets is homeomorphic to , the homeomorphism being . The map is called a covering map.
The definition may appear more intuitive if one imagines it this way: A continuous map is a covering map if each has an “evenly covered” neighbourhood! And what is “evenly covered”? It means that there the neighbourhood is such that the inverse image of that neighbourhood is just disjoint homeomorphic copies of the neighbourhood, and the homeomorphism is actually via the map . One can find images and examples on Wikipedia to get more intuition about this definition.
Now consider our figure-8 space and our Cayley graph . We define a continuous map in the following way. Map each of the tetravalent nodes of the Cayley graph to the centre and the attached edges to the appropriate circle.
The rendition of this map via a movement of the each point to its image is presented below.
Now looking at the above picture, it is a matter of concern to know if the map is truly continuous, as the smaller edges appear to be stretching too much. Indeed, this would not be the case if the Cayley graph is given the subspace topology of . In such a scenario, our map will fail to be continuous (I will leave it to the reader to verify this claim formally). Since the topology considered is that of a Simplicial 2-complex, the map can be constructed by glueing togeather continuous maps on each edge.
And as a matter of fact, we get more! The map is a covering map. This can be readily verified.
We said that if we have a covering map , then each has a “evenly covered” neighbourhood. We can think of the preimage of this “evenly covered” neighbourhood as a deck of cards. Informally speaking, a homeomorphism of that shuffles this deck around is called a deck transformation.
Formally, a deck transformation is a homeomorphism of the covering space such that .
Clearly, such a map will have to permute all the preimages of any point in via . For our very own covering map , we have the following two good deck transformations, again animated via a homotopy like the animation above.
In the image below, every tetravalent vertex on the main horizontal is mapped to the tetravalent vertex to its left. The animation runs the associated homotopy on an infinite loop.
Similar to the above image, each tetravalent vertex on the main vertical is mapped to the tetravalent vertex below it.
Seeing the two maps inspires to compose the two maps and get a more visually pleasing homeomorphism. Please forgive my skills for the unpleasant abruptness in the animation.
Composing the horizontal and vertical displacements (the first two images) and their inverses can give us many deck transformations of the covering space . In fact, a little more investigation would reveal that these are all the deck transformations. However, that investigation will take up a lot of explanation. An interested reader can find everything relevant in a book like Hatcher’s Algebraic Topology.
What was that visual we started with?
With the visual that was given in the beginning, I attempted to illustrate the property of the horizontal deck transformation of the covering space . The animation unloads the covering space from the image, applies the deck transformation and applies the covering map again. Therefore in the animation above, we can actually see the shuffling of preimages.
With this, I mark the end of this blog post. If you are able to spot any mistakes, please comment below to inform me about it. If you think some points need to be explained more clearly, comment below. If you have any opinions, remarks, criticisms, please write them below. Most of all, if you have any ideas for making an animation about something sufficiently mathematical, do communicate about it below!
Aside: Although not so useful a terminology topologically speaking, the term “Kissing circles” is sometimes used for two circles sharing a tangential point. This term was perhaps poetically introduced by Frederick Soddy in the context of Descartes’ theorem. You can find his piece of trigonometric romance here.