Let me state first off that I am a little uneasy posting to 
sci.physics.research since I understand so little of physics. But
a recent thread turned to Chern classes, which is not only
mathematics but a part of mathematics I understand moderately well,
so after a couple of email messages I have been persuaded to write in
with some explanatory material.

The question I can answer is, "what are Chern classes?" (Sorry, I can't
really help with the question, "why are they important to physics?")
The short answer is: they are cohomology classes associated to 
vector bundles over a topological space  M. I will try to outline a
few of their properties, which will be a little tricky since I am
not going to assume much familiarity with the terms "cohomology classes"
and "vector bundles", either.

Let me start by asking for a model of two topological spaces  E  and  B
and a two-to-one map  f: E-> B between them. Here are some examples:
 * E is a set with two points, B is a set with one points. No choice for  f !
 * E and B are both the unit circle in the plane, and f(z) = z^2
 * E and B are both cylinders, and  f  can be described as "rolling up
	the cylinder  E  twice as tightly to get the cylinder  B".

I intend the last example to sound like cheating after the second example.
If you recognize a cylinder as being a product  S^1 x R  of a circle and a 
line, you can describe the map I intended as  f(z,t) = (z^2, t).
In other words, we have a diagram of this shape:

                                E = circle
                                  |
                                  | squaring map
                                 \|/
                                  v
                     projection
        B' = cylinder ------>   B = circle

and we take advantage of it to get a two-to-one map of a cylinder  E'
onto  B'  sort of by using the "squaring" map on the right  and combining
it with the projection map on the bottom somehow. There is a trick to
doing this, which you don't really have to understand, but it works:
from the diagram above we construct another space  E'  and a mapping
E' --> B'  which, exactly as with the mapping on the right, is two-to-one.
This mapping  E' --> B'  is known as the "pull-back" of the squaring map
E --> B  (via the projection map on the bottom).

So we have this machine for making examples of two-to-one mappings:
start with one such -- say the squaring map I've been using -- and then
get oodles more by taking arbitrary maps into  B  and constructing
the pull-backs (whatever they are).

(OK, I can't resist giving the construction, because it's so easy.
Given any map  f : E --> B  between two spaces, and any other map
g : B' --> B  into the "base" space  B, we "pull back f via g" as
follows: let  E'  be the set of pairs  (b', e)  in the Cartesian product
B' x E  for which  g(b') = f(e)  (both sides are members of  B).
Let  f' : E' --> B'  be the projection onto the first component.
That's all there is to it.)

Now, there are plenty more examples of two-to-one maps. You may be 
familiar with the covering map  S^2 --> RP^2  which amounts to gluing
each point of the Southern hemisphere to its antipode in the Northern
hemisphere. This map isn't so easily related to the squaring map I
already mentioned. Well, actually, it is, but this new example is
more complicated -- you can derive the squaring-map example from this
new one, but not the other way around. What I mean is the observation
that there is a subspace  B  of  RP^2  which is homeomorphic to the
circle, and which is covered by the equator in  S^2, also a circle.
In other words, the squaring map is really just the pull-back, via the
inclusion  B --> RP^2,  of the covering map  S^2 --> RP^2.

And by the way, you can construct a pull-back of a pull-back; it's
another pull-back (exercise: what does this mean?). Therefore, all the
examples we could concoct starting with the squaring map may now
be created from the covering map  S^2 --> RP^2, too.

To recap: we had one example of a two-to-one map, we observed we could
get many more from this, but there were other examples which didn't
arise this way (trust me on this: the covering by S^2 does not arise
as a pullback of the squaring map). So we added another starting example,
then found that it and its pullbacks accounted for all the examples
known so far.

Are we done? Well, no: there's another example of a two-to-one map,
namely the covering  S^3 --> RP^3. It's not a pullback of the previous
example, although the previous one IS a pullback of this one! So this
one accounts for all the examples listed ... until we think about
the covering  S^4 --> RP^4, when the cycle repeats.

OK, this may seem like overkill, but it turns out you can lump together
all of these examples into one: there is a space called  S^\infty
(the "infinite dimensional sphere"), which includes all the spheres
S^1, S^2, ... (each is the equator of the next), and which covers 
a sphace known as  RP^\infty  in a two-to-one way. AND, at last, this
example is "universal": one can prove that ALL two-to-one maps
(in a certain category of topological spaces which is plenty general
enough for physicists :-) ) may be obtained as pullbacks of this
one covering  S^\infty --> RP^\infty, using all possible maps of
spaces into  RP^\infty.

This space we have now constructed,  RP^\infty, is then known as
the "classifying space for  Z/2Z  bundles"; the notation  B(Z/2Z)
is sometimes used ("B" as in "base space", see above). (With some 
categorical nonsense, it follows that once there is such an object, 
it's essentially unique. So I can use the word "the" with confidence...)

Now, you can sort of argue that  RP^\infty  is "made to order" for
two-to-one maps. I will press that idea in my next construction.
First, let me observe that  RP^n  can be thought of as the set of
all lines (through the origin) in  (n+1)-dimensional space. After all, 
every such line meets the sphere exactly twice at antipodal points,
so once you glue those pairs together, you have one point of  RP^n
for every line, and vice versa. This is sort of where all the 2's
come from: the group of isometries of the real line is  Z/2Z.
Or you can collect together all the lines, in the manner of the 
pullbacks described earlier: consider the set  E  of all the pairs  (r, v)  
where  r  is an element of  RP^\infinity (i.e. a line) and  v  is any
vector on that line. All the points in  E  with a given first coordinate
r  form a line, and indeed if you take a fairly small patch  U  of  RP^\infty,
all the pairs  (r,v)  in  E  whose first coordinate lies in the patch  U
will collectively form a subspace of  E  which is homeomorphic to the
product  U x R; we say the pair of space  E --> RP^\infty  is a "line bundle"
for this reason.


All right, now let's play this game with complex geometry.
There's a space called  CP^n  which consists of all the complex lines
(one-dimensional subspaces) in  C^(n+1). Each such line meets the
unit sphere in this space (which would be  S^(2n+1), if you're keeping
score); indeed, there's a circle's worth of points in each such
linear subspace at a distance of  1  from the origin. So you can
keep track of these 1-dimensional subspaces by taking  S^(2n+1)  and
sewing together these various circles together, collapsing each to
a point. If you're like me you have some trouble visualizing this, but
you needn't bother: all you have to know is that there is a
map  S^(2n+1) --> CP^n  with the property that it's a "circle-to-one" map
(the inverse image of each point in  CP^n  is a circle in  S^(2n+1) )
and that the points of  CP^n  correspond to complex lines. 

(Indeed, you can go whole hog and think of this map as simply being a 
restriction of the map  C^(n+1) - {0}  --> CP^n  which simply assigns a 
nonzero vector in  C^(n+1)  to the complex line which contains it. In the
language of equivalence relations, this is the "natural map"  v |--> [v],
but no one ever seems to think of the natural map as being natural
the first time they see it.)

The constructions for the various  n's  hang together nicely, and with
suitable topological niceties, one obtains a pair of space
S^\infty --> CP^\infty  which turn out to be "universal" for 
"circle-to-one" maps; you'll see  CP^\infty  referred to as  B(S^1)  or
B(U(1))  for this reason  (U(n)  is the unitary group. Oh yes, there are
some  n's coming...)

As in the real case, we may also form the total space  E  consisting
of all pairs  (r, v)  where  r  is an element of  CP^\infty, that is, a
complex line, and  v  is a vector in this line. Then  
E --> CP^\infty  is a complex line bundle: not only is the inverse
image of each point a copy of the complex line, but locally the inverse
images hang together (i.e., the inverse image of sufficiently small
neighborhoods  U  will be homeomorphic to products  U x C).
This space is universal for complex line bundles: in a suitable
category of spaces, every complex line bundle may be constructed
as a pullback of the bundle map  E --> CP^\infty  via some map into the
base space.

You can, of course, play all these games with higher-dimensional
subspaces of real or complex vector spaces than these simple one-dimensional
subspaces we've covered. Instead of projective spaces you get Grassmannian
manifolds, and instead of  Z/2Z  or  S^1  you get orthogonal groups or
unitary groups. Instead of line bundles you get vector bundles.
(You can even do it all over the quaternions if you're careful about
non-commutativity.)  The point is that all the bundles people worry about
can be assumed to be created once and for all in a universal way:
there exists a universal (real/complex) n-bundle  E --> B  for each  n,
and all the other bundles people concoct can be described simply
as pullbacks of this one universal one, pulled back along various maps
into the base space  B. It's sort of nice to be able to separate all
the bundle stuff into one giant concrete example, together with more
or less arbitrary maps into this scheme.

If you need to stop for a cup of coffee, now would be a good time.

Ok, welcome back. Now I'd like to talk about cohomology classes.
Cohomology is a very general term for a lot of very useful tools in
mathematics. The big plus is that they take some complicated (e.g.
geometrical) information and distill it into something algebraic.
One difficulty with the usual presentations, however, is that once
someone has computed a cohomology group, or a cohomology class,
they don't have a way of describing just what it is they have found.

Here is a way around it. From general categorical nonsense, one can
show that, whatever the homology group  H^n( X, G )  means for a 
topological space, it must be true that (with  n  and  G  fixed) there
is a space  K  for which  H^n(X, G)  is naturally isomorphic to the
set of (homotopy classes of) maps from  X  into  K. Here  G  is
assumed to be an abelian group, and then an abelian group structure
can be put on this set of maps, and the isomorphism is an isomorphism
of groups. To keep the various  K's  straight, we might write this as
K(G,n) rather than  K. It's called an Eilenberg-Maclane space.
If you grant that it exists, then some basic information about spheres
forces this space to have the property that all its homotopy groups
are zero except the  n-th, and that one is isomorphic to  G.
Almost none of these  K's  is a space you'd recognize unless you've
been keeping really disreputable company.

So that's what a "cohomology class" is (may be considered to be): it's
a (homotopy equivalence class of) map from the given space  X  into a
particular and peculiar space  K(G,n).

This can be a useful perspective for proving a few common facts about
cohomology. For example, suppose we have a map  f : X --> Y  between
two spaces. Given a cohomology class  u  in  H^n(Y,G), we think of
u  as a map  u : Y --> K(G,n); this of course leads to a composite map
(u o f) : X --> K(G,n), which is then an element of  H^n(X,G).
Typically we write   f^*(u)  instead of  u o f  but in any event it's
clear that  f  has created for us a function  f^* : H^n(Y,G) --> H^n(X,G).
It's not so clear but is true that  f^*  is actually a homomorphism of
groups, meaning that H^n( ---, G)  is a functor from the category of
spaces to the category of groups. (You may notice the change in direction
of the morphisms; to a mathematician this means it's a contravariant
functor, although I have been told that physicists use the words
covariant and contravariant differently so I'll just not call it anything.)

A little more yet is true (and not so accessible from this point of view):
if  G  happens to be (the underlying additive group of) a ring, then
the collection of groups  H^*(X,G)  also forms a ring, and  f^*  is a
homomorphism of rings. Since the most natural choices for  G  (namely
Z, the integers, and  R, the reals) are also rings, this fact is quite
handy. For example, one can give quite succint results this way:

Theorem: H^*( CP^\infty, Z ) = Z[x]  with  dim(x) = 2

This says that the cohomology ring is the ring of polynomials in one
variable. Then  H^n  is the subspace of homogeneous polynomials of 
dimension  n,  meaning the set of constant multiples of  x^(n/2)
(and if  n  is odd,  H^n  contains only  0 ). Additively, this is
just the group of integers, so we have simply asserted that  H^n  is
{0}  if  n  is odd, and  Z  if  n  is even.

The real case is a little more peculiar:

Theorem:  H^*(RP^\infty, Z) = Z[x]/(2x)  with  dim(x) = 1

(In other words, H^n  is  Z/2Z  for all n  except  n=0;  H^0  is  Z.)

Recall that  CP^\infty = B(U(1))  was to be just the first in a series
of spaces. I'll reveal the Secret for the next one and let you guess
the pattern for the others. (Your guess will be right :-) )

Theorem: H^*( B(U(2)), Z ) = Z[ c_1, c_2 ]  with  dim(c_n) = 2n.


So now we are able to reveal the Magic Decoder Ring for Chern Classes.

Suppose  E --> B  is a complex line bundle. 

By universality, there is a map  f: B --> CP^\infty  such that  E  is
simply the pullback of the universal line bundle  S^\infty --> CP^\infty.

By the first calculation (Theorem) above, H^*(CP^\infty, Z) = Z[c_1]
where  dim(c_1) = 2.

By functoriality there is a ring homomorphism  f^* : Z[c_1] --> H^*(B,Z).

By simple ring theory, this homomorphism is completely determined by
the image  f^*(c_1).

By quite a few homotopy-theory details I have glossed over :-) all of
this depends only on the line bundle   E-->B  with which we began.
Therefore we are justified in choosing a notation which involves
only this initial data. We write  c_1(E-->B)  (or  c_1(E)  or something)
for this cohomology class  f^*(c_1)  and call it the first Chern class
of the line bundle.

If we had begun with a vector bundle with higher dimension, we would
need to invoke a different classifying space, whose cohomology would
be a polynomial ring in several variables  c_i; we write  c_i(E)  for 
f^*(c_i), which is an element of  H^(2i)(B,Z).

If we had begun with a real vector bundle, we would be computing
elements  w_i(E) = f^*(w_i)  where  w_i  is an element of  H^i,
and is of additive order at most 2 (unless i=0); these are the 
Stiefel-Whitney classes.

I want to stress that many of the facts one might wish to prove about
Chern classes is now rather easy, or at least can be pushed away from
bundels altogether and reduced to certain calculations about the
classifying spaces. (Product formulas, action of Steenrod algebras, etc.)


Also I can't resist drawing the parallels:

	n-plane bundles over B are determined by maps into  B(U(n))

	elements of  H^n(B,Z)  are just maps into  K(Z,n)

This is particularly cool in the case  n=1  because  B(U(1))  is a  K(Z,2) !
(Also in the real case  B(O(1)) = K(Z/2Z, 1).)
Thus we find that line bundles are in one-to-one correspondence with
elements of  H^2(B,Z), and in fact it's the (first) Chern class which 
sets up this correspondence.