In the spirit of writing some sort of introductory posts to the things related to what I’m about to spend several years thinking and writing about, I thought I’d try to make a (more or less) layman friendly introduction to Homology and Homotopy.
It’s all residing in the realm of Topology. Topology is the field of mathematics, where those aspects of a shape not dependent on distances are studied. Thus rigidity is not interesting, whereas connectivity is. Narrow/thick is not interesting, but what kind of holes the surface has is. The ultimate thing to be said in topology about two objects is that they are homeomorphic, which technically means that there is an isomorphism between the objects in the category of topological spaces; and more comprehensibly means that there are continuous functions between the shapes such that they are each others inverses.
By a continuous function, we mean a function such that it respects the topology. By a topology, we take the collection of all possible open sets. (This makes for a generalisation of the usual “open intervals” and its ilk that makes things work out neatly)
Now, proving things to be homeomorphic is Really, Really Difficult. Proving things not to be homeomorphic isn’t really very easy either. Thus most mathematicians by now have given up on searching for direct proofs, and instead study invariants. If we can formulate some sort of associated entity in such a manner that whenever our topological spaces are homeomorphic, our entities are equal, then we can use that to conclude that whenever we have two spaces with different entities they can not possibly be homeomorphic.
For the sake of exposition I will stay within the realm of shapes within a plane for a while. Take for instance a circular disc and a circular ring – i.e. the disc with some subdisk removed. Intuitively, these are completely different shapes. But how would one show that they have to be? Here enters one of our first useful invariants. We can take paths within the space. Technically, these are functions from a closed intervals to the shape – but it’s easiest to just view them as curves traced with a pencil; or lengths of string positioned inside the shape. For a shape, we start out by simply picking a point anywhere on the shape, and look at curves that all begin and end in that point. We can form a new curve from any two such curves by simply continuing along the second once we’ve traced the first. Thus we get a composition of curves.
Next, we say that two curves are equivalent if we can nudge the lengths of string bit by bit, staying within the space all the time, until one lies on top of the other. If we look at all equivalent curves as one single entity, we still are able to multiply curves by putting one after the other. If we have a curve, and then continue along the same curve, but backwards, we get a shape that can be contracted along its length to vanish into the point at the beginning and end.
For those who have seen just a tiny bit of algebra, this is familiar. We have a set of objects – the curves – such that any two objects form a new object and there is some object that doesn’t change anything (adjoining the curve that doesn’t go anywhere but rather stays at the point we picked won’t change the curve we adjoin it to) and finally for any object there is some other object that ‘cancels’ it out. This is a group!
So we can attach a group to any topological space. This group is called the fundamental group and written π1. Now, by some thought, it can be shown that if the spaces are homeomorphic, then they will have isomorphic fundamental groups, so this group is one of the first topological invariants we find. There are several others that are easier to find – for instance a couple of connectivity attributes. If we just count the number of disjoint components our space decomposes into, it’s pretty obvious that if this number differs, then the spaces aren’t the same.
So… What is this π1 for different more or less wellknown things? For the disc and the circular ring above, we can calculate it rather easily. Within the disc, the strings can always be nudged anywhere – there isn’t anyhting within it that can obstruct such movements. So the fundamental group is trivial: any string can be retracted into just a point. On the ring, on the other hand, we can get stuck around the whole with our string. Furthermore, if we wind one string 2 times around the hole, and another 3 times around the hole, we cannot get these strings to each other. For an experiment, take a coffee cup, or some other nearby source of a ring shape with a single hole, and take a length of string, wind it 3 times through the hole, and tie it up to a loop. Now try to – without breaking the string or the knot – modify it until it only is winded 2 times. It’s impossible.
Thus, each element in the fundamental group of the circular ring can be characterised by the number of times we wind through, and in what direction. Any integer works, and by winding first 2 then 3 we end up winded 5=2+3 times, so in fact we can show this group to be isomorphic to , the group of integers under addition.
If we can do this for strings tied in loops, it should be possible to do similar things with say sheets tied into spheres or other higherdimensional similar things. The field dealing with this line of inquiry is called Homotopy – and has the fundamental problem that once we leave the fundamental group, things get REALLY messy. Not very much is known, even today, and research in the area is very active.
August 16, 2007: This post only generates spammy comments nowadays. I have disabled commenting on this post.