# Applications of diagram lemmas

I'm currently reading Theo Bühler's survey on exact categories about which he says

This article is written for the reader who wants to learn about exact categories and knows why. Very few motivating examples are given in this text

It turns out I don't really know any applications of homological algebra, so the article seems pretty dry to me. For example, every one seems to be very fond of diagram lemmas, but for the moment I see no interest in them. Is there any nice non-trivial theorem than can be proved with the tools developped in that text ? It would really help my reading become more pleasant.

EDIT: I'm very surprised to see that this question has not recieved any answer, or even any comments. I'm going to develop my question some more hoping that this changes.

While reading the article I always use the example of abelian groups. This helps me find interest in the article, since it gives a new (element free) perspective on familiar concepts, like the isomorphism theorems. However homological algebra seems to be a rather important theory, so I imagined there must be some theorem that can be formulated outside homological algebra and proved with its methods.

If this it not the case, then an answer saying so would perfectly answer my question. However I suspect this is not the case, since it seems homological algebra is connected to many areas of mathematics. If there are such theorems, but they require more heavy tools than those developed in the article, an answer explaining it would be very welcome.

• Algebraic topology uses homological algebra a lot. Take Brouwer's fixed point theorem, for example: no mention of anything even remotely close to homological algebra. But a standard proof uses (singular) homology, and to compute the homology of a sphere you use a long exact sequence... Boom, homological algebra appears. You want to compute homology with coefficients? Bam, derived functors "from nowhere". Homology of a product? Derived functors again. And insanity awaits you if you're interested in filtering spaces. – Najib Idrissi Jun 22 '15 at 11:44
• I think what you say about Brouwer's fixed point theorem is exactly what I need ! Do you happen to have a reference for that proof ? – Sergio Jun 22 '15 at 12:36

## 1 Answer

In general, algebraic topology uses homological algebra a lot. Historically, I believe homological algebra was first developed to study what happened in algebraic topology, and only split off as an independent area of study later.

Here is one example of an application of (very basic) homological algebra.

Theorem [Brouwer's fixed point theorem]. Every continuous map $f : D^n \to D^n$, where $D^n$ is the closed unit disk in $\mathbb{R}^n$, has a fixed point.

A sketch of proof goes like this (you can find more details at Wikipedia). Suppose you have a continuous $f : D^n \to D^n$ with no fixed point. You can use it to construct a retraction of $D^n$ onto $S^{n-1} = \partial D^n$, that is a map $g : D^n \to S^{n-1}$ such that $g(x) = x$ for $x \in S^{n-1}$: for $x \in D^n$, take the half-ray starting at $f(x)$ and passing through $x \neq f(x)$; call $g(x)$ the intersection of this half-ray with $S^{n-1}$.

But thanks to homology, such a retraction cannot exist, and it directly follows from the fact that $H_{n-1}(S^{n-1}) = \mathbb{Z}$ but $H_{n-1}(D^n) = 0$, but a retraction would induce a surjection $H_{n-1}(D^n) \to H_{n-1}(S^{n-1})$ by functoriality of homology. To compute $H_{n-1}(S^{n-1})$, one can repeatedly apply the Mayer–Vietoris theorem, and you see long exact sequences appearing. Granted, they are not very complicated long exact sequences (the only terms are either $0$ or $\mathbb{Z}$, and all the maps are either $0$ or an isomorphism), but just to show that these long exact sequences exist take a bit of homological algebraic theory already.

More generally, homological algebra is heavily used in algebraic topology.

Thanks to the Künneth theorem, computing the homology of a product reduces to compute the homology of each factor and of a Tor functor, a particular case of derived functor. So again homological algebra saves the day. To compute (co)homology of a space with different coefficients, one can use the Universal coefficient theorem, in which either a Tor or an Ext functor appears -- again, homological algebra.

Spectral sequences are a tool of choice in algebraic topology. As soon as you have a filtered space, you can use them to attempt to compute the homology of the whole space using the filtration, for example.
This has an enormous amount of applications; a first "simple" example be the Serre spectral sequence, which lets you compute the homology of the total space of a fibration using the base and the fiber (or the reverse, sometimes!). You can for example use that to compute the cohomology of $BU_n$ and find what are the characteristic classes of complex vector bundles -- something which doesn't have anything to do with homological algebra, a priori.
But to do computations with a spectral sequence, one usually needs a heavy amount of homological algebra technology. This is where many diagram lemmas are used; the snake lemma, the five lemma, the 3-3 lemma... are all very nifty tools one can use to reduce the amount of computation to do by hand.

• Thank you very much for your answer. I still can't make sense of most of it but I'm sure I soon will. – Sergio Jun 23 '15 at 10:16