The Unapologetic Mathematician

Exact Sequences

A sequence of groups is just a list of groups with homomorphisms going down the list: $...\rightarrow G_{n-1}\rightarrow G_n\rightarrow G_{n+1}\rightarrow ...$. We’ll use $d_n$ to refer to the homomorphism from $G_n$ to $G_{n+1}$. We say that a sequence is “exact” if the image of $d_{n-1}$ is the kernel of $d_n$ for each $n$.

What does this mean? First off, if we start with an element of $G_{n-1}$ and hit it with $d_{n-1}$ we get something in ${\rm Im}(d_{n-1})$. Since this is the same as ${\rm Ker}(d_n)$, if we now apply $d_n$ we go to the identity element of $G_{n+1}$. That is, the composition $d_n\circ d_{n-1}$ is always the trivial homomorphism sending all of $G_{n-1}$ to the identity in $G_{n+1}$.

On the other hand, if we have an element in the kernel of $d_n$ it’s also in the image of $d_{n-1}$. This means that if an element gets sent to the identity in the next group, it must have come from an element in the previous group.

So why should we care? Well, a number of different things can be very nicely said with exact sequences. If we write $\mathbf1$ for the group containing only one element, we can set up a sequence: ${\mathbf1}\rightarrow G\rightarrow^fH$. What does it mean for this sequence to be exact? Well there’s only one homomorphism from $\mathbf1$ to any group, and its image is just the identity element in $G$. So the kernel of $f$ is trivial — $f$ is a monomorphism.

Now let’s flip the diagram over to $G\rightarrow^fH\rightarrow{\mathbf1}$. There’s only one homomorphism possible from any group to $\mathbf1$, and its kernel is the whole domain. This means that the image of $f$ has to be all of $H$$f$ is an epimorphism.

Let’s put these two together to get a sequence ${\mathbf1}\rightarrow N\rightarrow^\iota G\rightarrow^\pi H\rightarrow{\mathbf1}$. Exactness at $N$ means that $\iota$ is a monomorphism, which we can think of as describing a copy of $N$ sitting inside $G$. Exactness at $H$ means that $\pi$ is an epimorphism. What does exactness at $G$ mean? The image of $\iota$ is that copy of $N$, which has to also be the kernel of $\pi$. That is, $H$ is (isomorphic to) $G/N$. We call any sequence of this form a “short exact sequence”.

Remember that the First Isomorphism Theorem tells us that we can factor any homomorphism into an epimorphism from the domain onto a quotient, followed by a monomorphism putting that quotient into the codomain. We can use that here to weave any exact sequence out of short exact sequences. Here is the (really cool) diagram:

Each of the diagonal lines is a short exact sequence, and as it says each (nontrivial) group off the main line is the image of one of the homomorphisms on the line and the kernel of the next.

We can also write an exact sequence ${\mathbf1}\rightarrow G\rightarrow H\rightarrow{\mathbf1}$. This just says that the homomorphism between $G$ and $H$ is an isomorphism. It’s really nice when this shows up in the middle of a longer exact sequence. If we can show that $G_{n-1}$ and $G_{n+2}$ are both trivial the sequence looks like $...\rightarrow{\mathbf1}\rightarrow G_n\rightarrow G_{n+1}\rightarrow{\mathbf1}\rightarrow ...$, so $G_n$ and $G_{n+1}$ are immediately isomorphic.

Another way exact sequences show up is in describing the structure of a group. We know that every group is a quotient of a free group. That is, there is some free group $F_1$ so that $F_1\rightarrow G\rightarrow{\mathbf1}$ is exact. Then the kernel of this projection is another group, so it’s the quotient of another free group $F_2$. Now the sequence $F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1}$ is exact. This is the presentation of $G$ by generators and relations. But the homomorphism from $F_2$ to $F_1$ might have a nontrivial kernel — there might be relations between the relations. In that case we can describe those relations as the quotent of another free group $F_3$: $F_3\rightarrow F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1}$ is exact. We can keep going like this to construct an exact sequence called a “free resolution of $G$“. It’s particularly nice if the process terminates at some point, giving a sequence ${\mathbf1}\rightarrow F_n\rightarrow F_{n-1}\rightarrow ...\rightarrow F_2\rightarrow F_1\rightarrow G\rightarrow{\mathbf1}$. A free resolution of a group that has only finitely many terms gives a lot of information about the structure of $G$.

March 6, 2007

History of Knot Theory

A friend of mine said offline that he’s looking forward to hearing more knot theory here. I’m looking forward to it too, especially as I get more of the algebraic basics down. I have a few posts I’m waiting until I get the chance to crank out some pictures, probably over spring break the next two weeks.

So I was excited to see today thar Jozef Przytycki has posted a chapter of his book on knots dealing with the history of the subject. It’s up on the arXiv.

Jozef was the first speaker I ever saw at an AMS meeting. The summer between my senior year of high school and my first year at the University of Maryland, they held a regional meeting in College Park. Since I’d already started getting into knot theory (more about that later), I sat in on the special session on knots, 3-manifolds, and their invariants. I was, to put it mildly, terrified. I understood nothing beyond the most basic terms.

I ran into a bunch of undergraduate students at the Joint Meetings back in January, and I remembered how it felt that first time. I’m sure it was small comfort, but I pointed out that just by being there, immersing themselves in the language, they were getting that step or two ahead of the game. After hearing the words flying around they’d wake up one day and just know something without really knowing where it came from. It’s like learning your native language: you don’t sit reading grammars, you’re immersed in it. You hear it all around and it just sinks in. Generally there’s also a lot of crying and messing yourself involved somewhere along the way, but you get past it and the language just feels natural.

One young student I remember in particular: an undergrad from Bard College named Tomasz Przytycki. Good luck, and remember to wipe.