Classifying Spaces Made Easy

John Baez

From week149 of This Week's Finds in Mathematical Physics

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 → ∞} [S^k(X), E(n+k)]

where S^k(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.

From week151 of This Week's Finds in Mathematical Physics

Now I want to say a bit more about the physics lurking in the space K(Z,2). I explained a bit about this space in "week149", but I've been pondering it a lot lately, so I'd like to say a bit more.

First let me review and elaborate on some basic stuff I said already. If G is any topological group, there is a topological space BG with a basepoint such that the space of loops in BG starting and ending at this point is homotopy equivalent to G. This space BG is unique up to homotopy equivalence.

BG is important because it's the "classifying space for G-bundles". What this means is that there's a principal G-bundle

p: EG → BG

called the "universal G-bundle", with the marvelous property that any principal G-bundle over any space X is a pullback of this one by some map

f: X → BG.

(I explained in "week149" how to pull back complex line bundles, and pulling back principal G-bundles works the same way.) Even better, two G-bundles that we get this way are isomorphic if and only if the maps they come from are homotopic! So there is a one-to-one correspondence between:

A) isomorphism classes of principal G-bundles over X

and

B) homotopy classes of maps from X to BG.

Now, suppose G is an abelian topological group. Then BG is better than a topological space with basepoint. It's an abelian topological group!

This means that we can iterate this trick. Starting with an abelian topological group G we can form BG, and BBG, and BBBG, and so on. This is called "delooping", because the loop space of each of these spaces is the previous one.

It's always fun to iterate any process whenever you can - Freud called this "repetition compulsion" - but there's more going on here than just that. In "week149" I said that when we have a list of spaces, each being the loop space of the previous one, it's called a "spectrum". And I said that we can use a spectrum to get a generalized cohomology theory. So we now have a trick for getting a generalized cohomology theory from a topological abelian group!

In particular, suppose we start with a plain old abelian group A. We can think of it as a topological group with the discrete topology - let's call this K(A,0). Then we can define

K(A,1) = B(K(A,0))

K(A,2) = B(K(A,1))

K(A,3) = B(K(A,2))

... and so on. We get a spectrum K(A,n) called an "Eilenberg-Mac Lane spectrum". The corresponding generalized cohomology theory is just ordinary cohomology with coeffients in the abelian group A! This means that

Hⁿ(X,A) = [X, K(A,n)]

where the right-hand side is the set of homotopy classes of maps from X to K(A,n). In short, K(A,n) knows everything there is to know about the nth cohomology with coefficients in A.

We've seen this trick a couple of times lately, and it's actually a big theme in homotopy theory: whenever we have some interesting invariant of spaces, we try to cook up a space that "represents" this invariant. I could say a LOT more about THIS idea, but that would propel us into further heights of abstraction, when what I really want is to come down to earth a bit. Just a little bit....

So: let's take A to be the integers, Z. As I said in "week149", we then get

K(Z,0) = Z,

K(Z,1) = U(1),

where U(1) is the group of "phases" or unit complex numbers, and

K(Z,2) = CP^∞

where CP^∞ is infinite-dimensional complex projective space. There are a couple of slightly different versions of this. Topologists like to start with the direct limit of the spaces Cⁿ, which they call C^∞. Then they take the space of all 1-dimensional subspaces and call that CP^∞. Mathematical physicists prefer to start with a Hilbert space of countable dimension. Then they take the space of unit vectors modulo phase. Both these versions are equally good models of K(Z,2). The first one is a lean, stripped-down version of the second.

Now U(1) is very important in quantum theory, and so are unit vectors modulo phase in a Hilbert space - physicists call these "pure states". So something cool is going on here. For some mysterious reason, it looks like K(Z,n)'s are important quantum physics! This is especially interesting because the abstract definition of the K(Z,n)'s has nothing to do with the complex numbers - just the integers. The complex numbers show up on their own accord. So maybe this hints at some explanation of why the complex numbers are important in quantum mechanics.

Why are K(Z,n)'s connected to quantum theory? I don't really know. But we can get some clues by asking some more specific questions.

First of all, why is K(Z,2) the same as CP^∞? In "week149" I just asserted this without proof. That's one of the fun things I'm allowed to do in this column. But let me sketch why it's true.

First I need to remind you of some more basic facts about topology. Suppose G is any topological group, and let P → X be any principal G-bundle. This gives us a long exact sequence of homotopy groups:

... → π_n+1(X) → π_n(G) → π_n(P) → π_n(X) → π_n-1(G) → ...

Two-thirds of the arrows in this sequence come from the maps

G → P → X

while the less obvious remaining one-third come from the map

LX → G

sending each loop in the base space to the holonomy of some connection on our bundle. Here LX means the space of based loops in X, and we're using the fact that

π_n(LX) = π_n+1(X)

which is obvious from the definition of the homotopy groups.

But now suppose P is contractible! Then all its homotopy groups vanish, so the above long exact sequence breaks up into lots of puny exact sequences like this:

0 → π_n+1(X) → π_n(G) → 0

or in other words:

0 → π_n(LX) → π_n(G) → 0

This says that the map from LX to G induces isomorphisms on all homotopy groups. By the Whitehead theorem, this implies that this map is a homotopy equivalence! So LX is really just G!! So X is just BG!!!

In short: if we have a space X with a principal G-bundle P over it, and P is contractible, X must be BG. [1]

Now let's use this fact to show that CP^∞ is K(Z,2). Remember that by our recursive definition,

K(Z,2) = B(K(Z,1)) = B(U(1))

so to show that CP^∞ is K(Z,2), we just need to find a principal U(1)-bundle over it with a contractible total space.

In "week149" we discussed a complex line bundle over CP^∞ called the "universal complex line bundle". If you take the space of unit vectors in a complex line bundle you get a principal U(1)-bundle. So let's do this to the universal complex line bundle. What do we get? We get a principal U(1)-bundle like this:

S^∞ → CP^∞

Being a mathematical physicist, I'm using S^∞ here to stand for the unit sphere in some countable-dimensional Hilbert space, and the map sends each unit vector to the corresponding pure state, or unit vector mod phase. Since there's a circle of unit vectors for each pure state, this is indeed a principal U(1)-bundle. But now for the cool part: the unit sphere in an infinite-dimensional Hilbert space is contractible! So we've got a principal U(1)-bundle with a contractible total space sitting over CP^∞, proving that CP^∞ is K(Z,2). Even better, the bundle

S^∞ → CP^∞

is the universal principal U(1)-bundle.

I can't resist explaining why the unit sphere in an infinite-dimensional Hilbert space is contractible. It seems very odd that a sphere could be contractible, but this is one of those funny things about infinite dimensions. Take our Hilbert space to be L²[0,1] and consider any function f in the unit sphere of this Hilbert space:

∫ |f(x)|² dx = 1

For t between 0 and 1, let f_t(x) be a function that equals 1 for x < t, and a sped-up version of f for x greater than or equal to t. If you do this right f_t will still lie in the unit sphere, and you'll have a way of contracting the whole unit sphere down to a single point, namely the constant function 1.

Cute, huh?

Next question: how does CP^∞ become an abelian topological group? There's a very pretty answer. Consider the space of rational functions of a single complex variable. This is a infinite-dimensional complex vector space, and there's a natural way to give it the topology of C^∞. This gives us a nice way to think of CP^∞: it's just the nonzero rational functions modulo multiplication by constants.

But nonzero rational functions form an abelian group under multiplication! And this is still true when we mod out by constant factors! So CP^∞ becomes an abelian group - and in fact an abelian topological group.

We can visualize CP^∞ quite easily this way. A rational function of a single complex variable has a bunch of zeros and poles - think of them as points on the Riemann sphere. We should really stick an integer at each of these points: a positive integer at each zero, and a negative integer at each pole, to tell us the order of that zero or pole. This gives enough information to completely specify the rational function up to a constant factor. So a point in CP^∞ is the same as a finite set of points on the sphere labelled by integers - which must add up to zero.

Of course, we have to get the right topology on CP^∞. As we move our point in CP^∞ around in a continuous way, the corresponding points on the sphere all move around continuously, like a swarm of flies... but when points collide, their numbers add! For example, when a point labelled by the number 7 collides with a point labelled by the number -3, it turns into a point labelled by the number 7 - 3 = 4.

In the lingo of physics, we've got a picture of points in CP^∞ as "collections of particles and antiparticles on the sphere". The integer at any point on the sphere tells us the number of particles sitting there - but if it's negative, it means we've got antiparticles there. Particle-antiparticle pairs can be created out of nothing, and they annihilate when they collide... it's very nice!

By the way, there's something called the Thom-Dold theorem that lets us generalize the heck out of this. We just showed that if you take the 2-sphere and consider the space of particle-antiparticle swarms in it, you get K(Z,2). But suppose instead we started with the n-sphere and considered the space of particle-antiparticle swarms in that. Then we'd get K(Z,n)!

More generally, suppose we didn't use integers to say how many particles were at each point in the n-sphere - suppose we used elements of some abelian group A. Then we'd get K(A,n)!

For more tricks like this, try this paper:

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

Now let me mention a different picture of K(Z,2), that's also nice, and also related to quantum theory. Take any countable-dimensional Hilbert space H and let U(H) be the group of unitary operators on H. Just like the unit sphere in this Hilbert space is contractible, it turns out that U(H) is contractible if we give it the norm topology or the strong topology.

Anyway, now let PU(H) be the "projective unitary group" of H, meaning the group of unitary operators modulo phase. There's an obvious map

U(H) → PU(H)

sending a circle's worth of points to each point in PU(H). It's easy to check that this is a principal U(1)-bundle. Since the total space U(H) is contractible, it follows that PU(H) is K(Z,2)!

This give a nonabelian group structure on K(Z,2), which may seem kind of weird, given that we just made it into an abelian group a minute ago. But I guess this other product is "abelian up to homotopy" in a very strong sense, so it's just as good as abelian for the purposes of homotopy theory.

Anyway, some people in Australia have figured out an extra trick you can do with this PU(H) group:

4) Alan L. Carey, Diarmuid Crowley and Michael K. Murray, Principal bundles and the Dixmier-Douady class, Comm. Math. Physics 193 (1998) 171-196, preprint available as hep-th/9702147.

Here's how it goes, at least in part. We say a linear operator

A: H → H

is "Hilbert-Schmidt" if the trace of AA* is finite. The space of Hilbert-Schmidt operators is a Hilbert space in its own right, with this inner product:

<A,B> = tr(AB*)

Let's call this Hilbert space X. U(H) acts on X by conjugation, and this gives an action of PU(H) on X, because phases commute with everything. This in turn gives an action of PU(H) on U(X)! Is your brain melting yet? Anyway, it turns out that this makes U(X) into the total space of a principal PU(H)-bundle:

PU(H) → U(X) → U(X)/PU(H)

But X is a countable-dimensional Hilbert space, so U(X) is contractible, so this is the universal principal PU(H)-bundle. And as we've seen, this means that

U(X)/PU(H) = B(PU(H))

but we just saw that

PU(H) = K(Z,2)

so

U(X)/PU(H) = B(PU(H)) = B(K(Z,2)) = K(Z,3) !

In "week149", I said I'd like K(Z,3) to be some sort of infinite- dimensional manifold closely related to quantum physics. I'm happier now, because here we are getting just that - technically, we're getting it to be a "Banach manifold". Of course, I could still complain that this description doesn't make the abelian group structure on K(Z,3) obvious. But it's definitely a big step towards understanding what K(Z,n)'s have to do with quantum theory.

While I'm at it, I should report some other things people have told me via email. If you ponder what I've said, you can see that CP^∞ has 2nd homology equal to Z, and that the generator of this homology group - the "universal cycle" - is given geometrically by the obvious way of sticking the sphere CP¹ inside CP^∞. This is nice because CP¹ is actually a submanifold of the manifold CP^∞. But according to email from Mark Goresky, Rene Thom has shown that for k > 6, we cannot make K(Z,n) into a manifold in such a way that the universal cycle is represented by a submanifold!

On the other hand, Michael Murray reports that Pawel Gajer has managed to make K(Z,n) into something called a "differential space", which is not quite a manifold, but good enough to do geometry on. I'm not sure how this relates to Thom's work... but anyway, I should read this stuff:

5) Pawel Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997), 155-207, also available as arXiv:alg-geom/9601025.

Pawel Gajer, Higher holonomies, geometric loop groups and smooth Deligne cohomology, Advances in Geometry, Birkhauser, Boston, 1999, pp. 195-235.

Now, so far I've been restraining myself from talking about "gerbes", but if you've gotten this far you must be pretty comfortable with abstract nonsense, so you'll probably like gerbes. Very roughly speaking, a gerbe is a categorified version of a principal bundle! Actually it's a categorified version of a sheaf, but sometimes we can think of it as analogous to the sheaf of sections of a bundle. And just as K(Z,2) is the classifying space for U(1) bundles, K(Z,3) is the classifying space for a certain sort of gerbe!

I sort of explained how this works in "week25", but you can read the details here:

6) Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhauser, Boston, 1993. ISBN 0-176-3644-7

What this means is that as we explore the meaning of these K(Z,n)'s for quantum theory, we are really categorifying familiar ideas from quantum theory. In particular, this story should keep going on forever: K(Z,4) should be the classifying space for a certain sort of 2-gerbe, and so on. But I don't think people have worked out the details beyond the case of 2-gerbes. If you want to learn about 2-gerbes, you have to read this:

7) Lawrence Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225, 1994.

Finally, for more applications to physics, try these papers:

8) Alan L. Carey and Michael K. Murray, Faddeev's anomaly and bundle gerbes, Lett. Math. Phys. 37 (1996), 29-36.

Jouko Mickelsson, Gerbes and Hamiltonian quantization of chiral fermions, Lie Theory and Its Applications in Physics, World Scientific, Singapore, 1996, pp. 216-225.

Michael K. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996), 403-416.

Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Index theory, gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707-722, preprint available as arXiv:hep-th/9511151.

Alan L. Carey, Michael K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories, J. Geom. Phys. 21 (1997) 183-197, preprint available as arXiv:hep-th/9511169.

Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Bundle gerbes applied to quantum field theory, Rev. Math. Phys. 12 (2000), 65-90, preprint available as arXiv:hep-th/9711133.

I thank N. Christopher Phillips of the University of Oregon, Michael K. Murray and Diarmuid Crowley of the University of Adelaide, and Mark Goresky of IHES for educating me about these matters... all remaining errors are mine!

Footnotes:

[1] Moreover, P must be the universal principal G-bundle. Conversely, for any topological group G the total space of the universal principal G-bundle is contractible. Everything fits together very neatly! But I don't need all this stuff now.

Examples of Classifying Spaces

The simplest nontrivial example is G = Z/2. This has the infinite-dimensional real projective space RP^∞ as its classifying space. It's nice to think of Z/2 as O(1), the 1×1 orthogonal real matrices - or in less pretentious terms, the real numbers of norm 1. So, we have

BO(1) = RP^∞

The reason this is nice is that we can replace the real numbers here by the complex numbers or the quaternions!

The group U(1) of 1×1 unitary complex matrices is just a pretentious name for the complex numbers of norm 1. This, in turn, is a pretentious name for the circle viewed as a group. The classifying space of this group is the infinite-dimensional complex projective space:

BU(1) = CP^∞

Similarly, the group Sp(1) of 1×1 unitary quaternionic matrices is just a pretentious name for the quaternions of norm 1 - which is a pretentious name for S³ viewed as a group. This group is also called SU(2), since it's isomorphic to the 2x2 unitary complex matrices with determinant 1. Anyway, the classifying space of this group is the infinite-dimensional quaternionic projective space:

BSp(1) = HP^∞

Of course it's no coincidence that the unit real numbers, unit complex numbers and unit quaternions are the only spheres that can be made into topological groups. So we see there's a nice description of BG whenever G is a sphere!

There's also a nice description of BG whenever G = S_n, the permutation group on n letters. Here EG is the space of all n-tuples of distinct points in an infinite-dimensional Hilbert space, with permutations acting in the obvious way. It's pretty easy to see that this space of n-tuples is contractible, and G acts freely on it, so it must be EG! The quotient space BG = EG/G is thus the space of all nordered n-tuples of distinct points in an infinite-dimensional Hilbert space.

You can also get nice descriptions of BG whenever G = Z/n. This generalizes the case of Z/2 in a nice way. Just as

B(Z/2) = S^∞ / (Z/2) = RP^∞

we have

B(Z/nZ) = S^∞ / (Z/n)

More precisely, let's think of S^∞ as the unit sphere in an infinite-dimensional complex Hilbert space, and let Z/n act on it as multiplication by the nth roots of unity. This infinite-dimensional sphere is contractible, and this action of Z/n is free, so everything is fine and dandy.

Also, whenever G is the fundamental group of a Riemann surface Σ of genus g where g = 1,2,3,..., the universal cover Σ^~ of this surface is contractible, and G acts freely on it, so we have EG = Σ^~ and BG = Σ. So, your friendly neighborhood two-holed torus is actually the classifying space of a group!

Also, don't forget BO(n), BU(n) and BSp(n)! These are Grassmannians: the space of all n-planes in an infinite-dimensional real, complex, or quaternionic Hilbert space. Not so easy to visualize - but they nicely generalize our earlier results for BO(1), BU(1) and Bp(1).

Finally, if G is an abelian topological group, so is BG, so you can iterate... and I've already given various descriptions of

BBB(Z) = BB(U(1)) = B(CP^∞)

This space is also called K(Z,3), since we took Z and hit it with "B" thrice.

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