Homology

It's been a while since I gave an update on my research project, and since I got working on blogger, it's time to write a new post.

So to explain what Smith Normal Form is, I should explain what it does, and to do that I need to explain the problems that I am looking at. I'm going to assume that you are at least somewhat familiar with the concept of a group and quotient groups. If not, read the wikipedia article on it.

So the first important idea is a homology group. Suppose that we have an infinite sequence of abelian groups and homomorphisms that satisfy the condition (that is, applying two of these homomorphisms always results in the identity). Then we see immediately that . Furthermore, since these are abelian groups, we have , so the quotient group is defined. This infinite sequence of abelian groups is called a chain complex and these quotient groups are called the homology groups. The nth homology group is defined as .

Okay, that's probably pretty opaque if you don't already know what I'm talking about, so here is a simple example (which will need an example of its own to illustrate, but that will come in time) called simplicial homology. We will build up a space as follows: To begin, we start with some set of isolated points, which we will call the 0-skeleton of our space. Next, we'll draw some lines (these are topological lines, so we're assuming that none of them touch except at points in the 0-skeleton and they don't have to be straight) between points of the 0-skeleton. The collection of points and lines that results will be called the 1-skeleton. Now, we take some triangles (that are "filled in") and glue them on so that their boundary is part of the 1-skeleton. The 1-skeleton, along with these triangles, form the 2-skeleton. We create the 3-skeleton, the 4-skeleton, et cetera by gluing in tetrahedra, 4-simplices, 5-simplices, and so on. (this gets harder to visualize past the 2-skeleton, so I'll stop there).

Now let's define a chain complex on this simplicial complex as follows. Our abelian groups are the free abelian groups generated by the n-simplices (so they'll all be isomorphic to for some k). The homomorphisms are the boundary maps, with orientation done so that the composition requirement is satisfied. For example, the boundary of a line will be (as one possibility), the formal difference of the start point and the end point (the reason for using addition as the group operation here will become apparent later). Then the boundary of a triangle will be directed so that the differences all cancel out when the boundary is taken again.

So as an example consider the following simplicial complex:
This complex is made up of six points, nine lines, and three triangles. So the chain groups are , and all the rest are trivial. We can then compute the homology groups. The first homology group is . I claim that this is isomorphic to . To see this, consider the top triangle. Suppose that our element of has a component with the top right line. Then the boundary includes a contribution of the top point. We must then have an equal component of the top left line to cancel out this contribution, but we have the relation that the sum of the boundary of the top triangle is trivial (because of the quotient). That means that we can effectively replace all the components of the top left and top right lines with the bottom line of the top triangle. Similarly, we can do this for the other two triangles and reduce our element to just a combination of the middle three lines. Then for these to have trivial boundary, the three components must all be equal, but they can be anything. This means that our homology group is , since it is simply determined by how many times it contains this loop around the center. The second homology group is much simpler, since the image of is trivial because is trivial, so the second homology group is the kernel of . The kernel of is also trivial, since the boundaries of the three triangles are all independent and none of them are empty. This means that the second homology group is the trivial group.

Wooo, if you followed that you probably understand as much as I do about homology (although I was pretty lazy about keeping track of orientation, which is bad)! In the next post on this series I'll explain what this has to do with matrices and Smith normal form.