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 h^{n}(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

h^{n}(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 h^{n}(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, S^{1}.
You can check that the space of based
loops in S^{1} 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 CP^{n}, but you can define CP^{∞}
as a "direct limit" of these
as n approaches infinity, using the fact that CP^{n} sits inside
CP^{n+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)] = H^{2}(X)

where H^{2}(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 H
^{2}(X).

So now you know this: if you hand me a complex line bundle over X,
I can cook up an element of H^{2}(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 H^{2}(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 S^{2}. Since

H^{2}(S^{2}) = Z

this means that first Chern class of a line bundle over
S^{2} 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:

H^{2}(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^{*}: H^{2}(CP^{∞}) → H^{2}(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, H^{2}(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

H^{2}(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 → CPand then take the pullback of L by fg.^{∞}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
H^{n}(X) even when n is negative, via:

H^{n}(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.

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

H^{n}(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 C^{n}, 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 L^{2}[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.

For more tricks like this, try this paper:

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

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 CP^{1} inside CP^{∞}. This is nice
because CP^{1} 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.

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 S^{3} 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.

© 2009 John Baez

baez@math.removethis.ucr.andthis.edu