Generalized Cohomology Theories and Eilenberg-Mac Lane Spaces

So: what's a "generalized cohomology theory"?

This is a gadget that eats a topological space X and spits out a sequence of abelian groups hn(X). To be a generalized cohomology theory, this gadget must satisfy a bunch of axioms called the Eilenberg-Steenrod axioms. The most basic example is so-called ordinary cohomology, so when you're first learning this stuff the main motivation for the Eilenberg-Steenrod axioms is that they're all satisfied by ordinary cohomology. But there are lots of other examples: various flavors of K-theory, cobordism theory, and so on. Eventually, you learn that underlying any generalized cohomology theory there is a list of spaces E(n) such that

hn(X) = [X, E(n)]

where the right-hand side is the set of homotopy classes of maps from X to E(n). We say this list of spaces E(n) "represents" the generalized cohomology theory. Moreover, these spaces fit together to form a "spectrum", meaning that the space of based loops in E(n) is E(n-1). It follows that each space E(n) is an infinite loop space: a space of loops in a space of loops in a space of loops in... where you can go on as far as you like.

Conversely, given an infinite loop space E(0), we can use it to cook up a spectrum and thus a generalized cohomology theory. So generalized cohomology theories, spectra and infinite loop spaces are almost the same thing.

But what's so important about them?

Well, secretly an infinite loop space is nothing but a homotopy theorist's version of an abelian group. A bit more technically, we could call it a "homotopy coherent abelian group". By this I mean a space with a continuous binary operation satisfying all the usual laws for an abelian group up to homotopy, where these homotopies satisfy all the nice laws you can imagine up to homotopy, and so on ad infinitum. In the context of homotopy theory, this is almost as good as an abelian group. Pretty much anything a normal mathematician can do with an abelian group, a homotopy theorist can do with an infinite loop space!

For example, normal mathematicians often like to take an abelian group and equip it with an extra operation called "multiplication" that makes it into a ring. Homotopy theorists like to do the same for infinite loop spaces. But of course, the homotopy theorists only demand that the ring axioms hold up to homotopy, where the homotopies satisfy a bunch of nice laws up to homotopy, and so on. Usually they do this in the context of spectra rather than infinite loop spaces - a distinction too technical for me to worry about here! - so they call this sort of thing a "ring spectrum". Similarly, corresponding to a commutative ring, the homotopy theorists have a notion called an "E ∞ ring spectrum". The word "E ∞ " is just a funny way of saying that the commutative law holds up to homotopy, with the homotopies satisfying a bunch of laws up to homotopy, etcetera.

If you start with a ring spectrum, the corresponding cohomology theory will have products. In other words, the cohomology groups hn(X) of any space X will fit together to form a graded ring called h*(X) - the star stands for a little blank where you can stick in any number "n". And if your ring spectrum is an E ∞ ring spectrum, h*(X) will be graded-commutative. This is what happens in most of really famous generalized cohomology theories. For example, the ordinary cohomology of a space is actually a graded-commutative ring with a product called the "cap product", and similar things are true for the most popular flavors of K-theory and cobordism theory.

Of course, it's quite a bit of work to make all this stuff precise: people spent a lot of energy on it back in the 1970's. But it's very beautiful, so everybody should learn it. For the details, try:

1) J. Adams, Infinite Loop Spaces, Princeton U. Press, Princeton, 1978.

2) J. Adams, Stable Homotopy and Generalized Homology, Chicago Lectures in Mathematics, U. Chicago Press, Chicago, 1974.

3) J. P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics 271, Springer Verlag, Berlin, 1972.

4) J. P. May, F. Quinn, N. Ray and J. Tornehave, E ∞ Ring Spaces and E ∞ Ring Spectra, Lecture Notes in Mathematics 577, Springer Verlag, Berlin, 1977.

5) G. Carlsson and R. Milgram, Stable homotopy and iterated loop spaces, in Handbook of Algebraic Topology, ed. I. M. James, North-Holland, 1995.

Now, there's a particularly nice class of generalized cohomology theories called "complex oriented cohomology theories". Elliptic cohomology is one of these, so to understand elliptic cohomology you first have to study these guys a bit. Instead of just giving you the definition, I'll lead up to it rather gradually....

Let's start with the integers, Z. These form an abelian group under addition, so by what I said above they are a pitifully simple special case of an infinite loop space. So there's some space with a basepoint called K(Z,1) such that the space of all based loops in K(Z,1) is Z.

Be careful here: I'm now using the word "is" the way homotopy theorists do! I really mean the space of based loops in K(Z,1) is homotopy equivalent to Z. But since we're doing homotopy theory, that's good enough.

Okay: so there's a space K(Z,1) such that the space of all based loops in K(Z,1) is Z. Similarly, there's a space K(Z,2) such that the space of all based loops in K(Z,2) is K(Z,1). And so on... that's what it means to say that Z is an infinite loop space.

These spaces K(Z,n) are called "Eilenberg-Mac Lane spaces", and they fit together to form a spectrum called the Eilenberg-Mac Lane spectrum. Since it's built using only the integers, this is the simplest, nicest spectrum in the world. Thus the generalized cohomology theory it represents has got to be something simple and nice. And it is: it's just ordinary cohomology!

But what do the spaces K(Z,n) actually look like?

Well, for starters, K(Z,0) is just Z, by definition.

K(Z,1) is just the circle, S1. You can check that the space of based loops in S1 is homotopy equivalent to Z - the key is that such loops are classified up to homotopy by an integer called the winding number. In quantum physics, K(Z,1) usually goes by the name U(1) - the group of unit complex numbers, or "phases".

K(Z,2) is a bit more complicated: it's infinite-dimensional complex projective space, CP∞! I talked a bunch about projective spaces in "week106". There I only talked about finite-dimensional ones like CPn, but you can define CP∞ as a "direct limit" of these as n approaches infinity, using the fact that CPn sits inside CPn+1 as a subspace. Alternatively, you can take your favorite complex Hilbert space H with countably infinite dimension and form the space of all 1-dimensional subspaces in H. This gives a slightly fatter version of CP∞, but it's homotopy equivalent, and it's a very natural thing to study if you're a physicist: it's just the space of all "pure states" of the quantum system whose Hilbert space is H.

How about K(Z,3)? Well, I don't know a nice geometrical description of this one. And this really pisses me off! There should be some nice way to think of K(Z,3) as some sort of infinite-dimensional manifold. What is it? Does anyone know? Jean-Luc Brylinski raised this question at the Conference on Higher Category Theory and Physics in 1997, and it's been bugging me ever since. From the work of Brylinski which I summarized in "week25", it's clear that a good answer should shed light on stuff like quantum theory and string theory. Basically, the point is that the integers, the group U(1), and infinite-dimensional complex projective space are all really important in quantum theory. This is perhaps more obvious for the latter two spaces - the integers are so basic that it's hard to see what's so "quantum-mechanical" about them. However, since each of these spaces is just the loop space of the next, they're all part of tightly linked sequence... and I want to know what comes next!

But I'm digressing. I really want to focus on K(Z,2), or in other words, infinite-dimensional complex projective space. Note that there's an obvious complex line bundle over this space. Remember, each point in CP∞ is really a 1-dimensional subspace in some Hilbert space H. So we can use these 1-dimensional subspaces as the fibers of a complex line bundle over CP∞, called the "canonical bundle". I'll call this line bundle L.

The complex line bundle L is important because it's "universal": all the rest can be obtained from this one! More precisely, suppose we have any topological space X and any map

f: X → CP∞

Then we can form a complex line bundle over X whose fiber over any point x is just the fiber of L over the point f(x). This bundle is called the "pullback" of L by the map f. And the really cool part is that any complex line bundle over any space X is isomorphic to the pullback of L by some map! Even better, two such line bundles are isomorphic if and only if the maps f defining them are homotopic! This reduces the study of many questions about complex line bundles to the study of this guy L.

For example, suppose we want to classify complex line bundles over any space X. From what I just said, this task is equivalent to the task of classifying homotopy classes of maps

f: X → CP∞.

But remember, CP∞ is the Eilenberg-Maclane space K(Z,2), and the Eilenberg-Maclane spectrum represents ordinary cohomology! So

[X, CP∞] = [X, K(Z,2)] = H2(X)

where H2(X) stands for the 2nd ordinary cohomology group of X. So the following things are really the same:

isomorphism classes of complex line bundles over X

homotopy classes of maps from X to CP ∞

elements of the ordinary cohomology group H2(X).

So now you know this: if you hand me a complex line bundle over X, I can cook up an element of H2(X). People call this the "first Chern class" of the line bundle. If you hand me two complex line bundles, I can tell if they're isomorphic by seeing if their first Chern classes are equal. Conversely, if you hand me any element of H2(X), I can cook up a complex line bundle over X whose first Chern class is that element.

Of course, I haven't really explained how I cook up all these things. To learn that, you need to study this stuff a bit more.

But let's consider a couple of examples. Suppose X is the 2-sphere S2. Since

H2(S2) = Z

this means that first Chern class of a line bundle over S2 is secretly just an integer. People call this the "first Chern number" of the line bundle. The first physicist to get excited about this was Dirac, who bumped into this idea when thinking about magnetic monopoles and charge quantization. Dirac didn't know about complex line bundles and Chern classes - he was just studying the change of phase of an electrically charged particle as you move it around in the magnetic field produced by a monopole! But later, the physicist Yang met the mathematician Chern and translated Dirac's work into the language of line bundles. See

6) C. N. Yang, Magnetic monopoles, fiber bundles and gauge field, in Selected Papers, 1945--1980, with Commentary, W. H. Freeman and Company, San Francisco, 1983.

for the full story.

Next let's try a curiously self-referential example. It should be fun to classify complex line bundles on CP∞, since this is where the universal one lives! So let's take X = CP∞. Since CP∞ is K(Z,2), a little abstract nonsense shows that it's ordinary 2nd cohomology group is Z:

H2(CP∞) = [CP∞, CP∞] = Z.

This means that the first Chern class of a complex line bundle over CP∞ is secretly just an integer. But what's the first Chern class of the universal complex line bundle, L? Well, this bundle is the pullback of itself via the identity map

1: CP∞ → CP∞

and this map corresponds to the element 1 in [CP∞, CP∞] = Z. So the first Chern class of L is 1. See how tautologous this argument is? It sounds like it's saying something profound, but once you understand it, it's really just saying 1 = 1.

The first Chern class of the universal bundle L is really important, so let's call it c. It's important because it's universal: it gives us a nice way to think of the first Chern class of any complex line bundle. Up to isomorphism, any complex line bundle over any space X comes from some map

f: X → CP∞

so to compute the first Chern class of this line bundle, we can just work out f*(c), where

f*: H2(CP∞) → H2(X)

is the map induced by f. If you don't see why this is true, think about it a while - it's just a big fat tautology!

The ideas we've been discussing raise some obvious questions. For example, H2(X) isn't just a set: it's an abelian group. We already knew this from our basic course in algebraic topology, and now we also know another explanation: CP∞ is an infinite loop space, so it's like an abelian group for the purposes of homotopy theory. In fact, this particular infinite loop space actually is an abelian group. Maps from anything into an abelian group form an abelian group, which makes

H2(X) = [X, CP∞]

into an abelian group. But now you're dying to know: what exactly do the product map

m: CP∞ x CP∞ → CP∞

and the inverse map

i: CP∞ → CP∞

look like? And what does all this mean for the set of isomorphism classes of complex line bundles on X? It's an abelian group - but what are products and inverses like in this abelian group?

Well, I won't answer the first question here: there's a very nice explicit answer, and you can describe it in terms of particles and antiparticles running around on the Riemann sphere, but it would be too much of a digression to talk about it here. To learn more, study the "Thom-Dold theorem" and also some stuff about "configuration spaces" in topology:

7) Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91-107.

The second question is much easier: the set of isomorphism classes of complex line bundles on a space X becomes an abelian group with tensor product of line bundles as the product. Taking the dual of a line bundle gives the inverse in this group.

Putting these ideas together, we get a nice description of tensoring line bundles in terms of the product

m: CP∞ x CP∞ → CP∞

which I can explain even without saying what the product looks like. Suppose I have two line bundles on X and I want to tensor them. I might as well assume they are pullbacks of the universal bundle L by some maps

f: X → CP∞,

g: X → CP∞.

It follows from what we've seen that to tensor these bundles, I can just form the map

fg: X → CP∞

given as the composite

(f,g) m X → CP∞ x CP∞ → CP∞

In other words: since the canonical line bundle on CP∞ is universal, CP∞ knows everything there is to know about complex line bundles. In particular, it knows everything there is to know about tensoring complex line bundles: the operation of tensoring is encoded in the product on CP∞. Similarly, the operation of taking the dual of a complex line bundle is encoded in the inverse operation

i: CP∞ → CP∞.

Footnote:

[1] Almost, but not quite: if I hand you the infinite loop space E(0), you can only recover one connected component of the infinite loop space E(1), namely the component containing the basepoint. So there is more information in a spectrum than there is in an infinite loop space. A spectrum is a sequence of infinite loop spaces where the based loops in E(n) form the space E(n-1); starting from a single infinite loop space we can cook up a spectrum, but it will be a spectrum of a special sort, called a "connective" spectrum, where the spaces E(n) are connected for n > 0.

Given a spectrum we can define the generalized cohomology groups Hn(X) even when n is negative, via:

Hn(X) = lim k → ∞ [Sk(X), E(n+k)]

where Sk(X) denotes the k-fold suspension of X. If the spectrum is connective, these groups will vanish when n is negative. A good example of a connective spectrum is the spectrum for ordinary cohomology (the Eilenberg-Mac Lane spectrum). A good example of a nonconnective spectrum is the spectrum for real or complex K-theory.



