Robert's Mistakes

Cohomology for kids, part 1: introduction

Dec 24, 2019

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century.

What a grandiose thing to say! And it's true! Keep reading!

Here's this post on twitter: [twitter].

Homology is a topological invariant

To any given space, let's call it MM as in manifold, a homology associates some data: a tower of homology groups Hi(M)H_i(M), one for every dimension ii. For example:

H0(circle)=ZH1(circle)=ZH2(circle)=0...Hi(circle)=0,for all other i H_0(circle) = \Z \\ H_1(circle) = \Z \\ H_2(circle) = 0 \\ ... \\ H_i(circle) = 0, \textnormal{for all other i}

That Z\Z symbol indeed means the integers, but, for now, only as a group without multiplication and merely addition as operation. In fact, Z\Z is also a parameter to the homology, and I should have written Hi(M,Z)H_i(M,\Z) instead. But through most of our discussion we will stick to Z\Z-homologies, as they're both: simple and especially interesting. In this setting Z\Z is the simplest building block for homology groups, and corresponds to a single "cycle". This means that, in some sense, a circle is the most basic object to compute a homology for.

Homology is an invariant in the sense that, if you want to know wether two spaces are the same, topologically speaking, you can begin by comparing their homology groups. If these differ, then the spaces must be different. If they have the same homology groups, however, then they may be the same, or not! So the conclusive arrow goes this way:

topology the same    invariants the same \textnormal{topology the same} \implies \textnormal{invariants the same}

But if you do classical simplical homology, that is, you're using nets of simplices on manifolds, or singular homology, a slight generalization where simplical nets that may not admit a corresponding manifold are also allowed, then there's only one more piece of information missing to completely classify topological nets: torsion groups! With them, the link also holds the other way around:

topology the same    invariants the same \textnormal{topology the same} \impliedby \textnormal{invariants the same}

Not all homology theories have torsion groups, but I will explain them anyways, because it's good to know about torsion!

But first: invariant in what sense? What does it mean, that two topological spaces are essentially the same? Since we're doing topology the best answer would be homotopy. To prove that something is a topological invariant we'd have to show that it doesn't change when we apply a homotopy. In fact, homotopy itself can be generalized to a topological invariant, which serves nicely to motivate homology, whilst lighting up some basics. Let's have that first:

Homotopy groups

Take a manifold, or surface. Think of a path p1p_1 on that surface, say, connecting a pair of points A,BA, B. Now, think of another path p2p_2, also connecting AA and BB. We say p1p_1 and p2p_2 are homotopic, if one can be deformed smoothly into the other, completely within the surface. Whithout losing much generality we can simplify this picture a little, assume our manifold is connected, and limit our discussion to loops attached to a single chosen point, and then ask if two such loops are homotopic.

This can fail, for example, on a torus, where there exists a class of loops that can be shrunk to a point, but there are also two other kinds of loops which cannot: one following the hoop and homotopically equivalent to the "equator", and a "polar" class of loops going through the hole.

Actually, there are many inequivalent loops of the through the hole kind: those going around once, or twice, or three times... or even backwards, the other way around! Paths should carry little arrow heads to make that distinction possible. We might combine a pair of loops by cuttung them near our base point, and then reconnecting across to make a single, longer loop.

Let's call that operation adding a pair of loops. Because all this is awfully similar to the integers with addition. You should check that "adding" a +1+1 and a 1-1 loop along the same once-around loop produces a "trivial" zero-loop: which can then be contracted to a point.

So let's call those once-around loops cycles. So we could say that each cycle gives a copy of Z\Z. On a torus there are two cycles AA and BB and the set of all its loop classes behaves exactly like the free group over two generators: We can label its elements using strings of powers of A and B, with symbols alternating like so:

AB2A1B AB^2A^{-1}B

We use power notation here, because it saves space which makes it look better. Using a polynomial instead would suggest that the group were commutative, which isn't true for homotopy groups. Now you go and have a look at the entirely delightful actual definition of homotopy, which underpins the basic idea I just outlined with a nice and smooth parametrization for paths, enabling you to do this stuff on actual geometric data:

If you did that, you should see how to generalize the idea higher dimensions: by blending one homotopy into another. Paths-between-paths-between-paths! The next higher dimensional analog for our cycle would be a sphere, and a multiple "loop" would have to wrap another sphere around it multiple times somehow...

π0(sphere)=Zπ1(sphere)=0π2(sphere)=Zπ3(sphere)=Zπ4(sphere)=Z2... \pi_0(sphere) = \Z \\ \pi_1(sphere) = 0 \\ \pi_2(sphere) = \Z \\ \pi_3(sphere) = \Z \\ \pi_4(sphere) = \Z_2 \\ ... \\

Homotopy groups for spheres don't vanish for i>ni > n, and quickly things become much more complicated. See here for more:

You find this difficult to visualize? It turns out, even just computing homotopy groups is very difficult in general. And that's why Henri Poincare invented a commutative "abelian" alternative: homology!

If a string of powers of n symbols is the free group over n generators, then its abelianization is an orthogonal lattice in n dimensions. Naming the number of times each symbol appears is enough, we don't need to specify strings.

Clearly the ideas were in the air before, but it was Poincare who fused them all into a coherent picture. And his early concept already included torsion! Let's take a look at the beautiful mechanism with which we can compute Z-homologies:

Simplex and CW-Complex

A simplex is a generalized triangle. We will need a bunch of those to cover our manifolds, because we are about to generalize the well-known technique of triangulating a surface, so we can deal with higher-dimensional n-volumes in pretty much the same way.

Anything you can make from a bunch of simplices, by gluing matching boundaries together, can be called a simplical complex. If other polytopes appear, the name is CW complex. Which is what we will be holding on to.

Introduced by John Henry Constantine Whitehead for the explicit purpose of doing homology, CW-complexes generalize simplical complexes (see also simplical homology) in that they dispense with the requirement that the result should correspond to a sensible surface.

Generalizing further, a simplical set is the same as purely algebraic artifact, which as such does not carry any topological meaning. So yes, we can play on graphs, or have an antenna pop out from a polyhedron, and yet be able to compute homology groups. And that is just a small part of the dominating feature of the mathematical landscape I claimed it became. But I digress...

Where do these strange words come from?

CW stands for 'closure-finite' and 'weak' topology. And while we're at it, Ben Steffan explained:

The "singular" in the name [singular homology] refers to the fact that the maps you study are only required to be continuous and can thus exhibit "singularities" and pathological behavior. [twitter]

And David Roberts knew why it's called a 'complex':

I said it was due to lots of things being called 'complexes' in the [19]30s and 40s, as they were all made up of many parts. As the subject matured, and was indeed fully axiomatised once categorical language arrived, we were stuck with the old names. [twitter]

It's basically Carl Friedrich Gauss' fault! Mike Stay said about that:

Gauss was first to use the term "complex number" in an 1831 article about a book he'd just published. He spends some time explaining the geometry of complex numbers to show how they have many parts (e.g. "a complex" in English) but aren't complicated. [ref] [twitter]

And because Gauss was paragon, it didn't take too long until calling things 'complex' became fashionable. I should tell you more about history, later.

Puzzle: I don't know about the origin of the words homotopy, or homology. If you do, please tell, I'd love to know!