# Characteristic Classes
Let be a complex vector bundle. Choose a Hermitian structure on and let denote its bundle of unitary frames. So a point of consists of a pair where and is an orthonormal basis of . The group acts on the right, where
This makes into a principal -bundle over and hence we get a map
The elements of the images of this map gives a subring of the cohomology ring of called the ring of characteristic classes of the vector bundle . Its definition depended on the choice of Hermitian metric, but it can be shown that does not depend on this choice, justifying the terminology "characteristic".
In the above, we could have taken to be a real vector bundle, a real scalar product, and to be the bundle of orthonormal frames. The group would then be the orthonormal group . Similarly, we could have taken to be an oriented real frame bundle, a real scalar product, to be the bundle of oriented orthonormal frames, and the special orthogonal group.
In all three cases, , and we must examine the ring whose structure we shall investigate below. In each case there are standard generators, whose images under are called Chern classes for the case of , Pontryagin classes for the case of , and one class, called the Pfaffian, in addition to the Pontrygin classes in the case of .
# The Invariants
We can identify the Lie algebra of with the space of all matrices of the form , where is self-adjoint, and hence (using the trace pairing and forgetting the unneeded ) may identify with the space of self-adjoint matrices with the coadjoint action being conjugation:
Define the polynomial of degree in to be the coefficient of in the characteristic polynomial of :
For instance, and . The polynomials are clearly invariant under the adjoint representation, since the characteristic polynomial is. It is a theorem that they generate the ring on invariants.
The characteristic classes corresponding to the for a complex vector bundle are called its Chern classes.
We may identify and also with the space of skew-adjoint matrices. For such a matric we have
so all the coefficient of in the characteristic polynomial of vanish when is odd. We may write
The polynomials of degree generate the ring . The corresponding characteristic classes for a real vector bundle are called its Pontryagin classes.
The Lie algebra is the same for so all the are invariant polynomials. There is one additional invariant which is not a polynomial in the , called the Pfaffian. It is defined in the following way: To each and , set
where denotes the scalar product. We have
so is an alternating bilinear form and the map is a linear isomorphism. The element
is an element of and the map depends only on the scalar product and so it invariant. However the group preserves a basis element, , of , where
Here is any oriented orthonormgal basis and is the dual basis. We may then define by
It is a polynomial function of of degree which is invariant. For any we can find an oriented orthonormal basis relative to which the matrix takes the form
Relative to this orthonormal basis we have
On the other hand,
so we have the general formula
So the square of is equal to . Also note that in odd dimensions, the determinant of any antisymmetric matrix vanishes, so this phenomenon does not occur.
The characteristic class corresponding to the Pfaffian for a real, -dimensional, oriented vector bundle is called the Euler class of the vector bundle, and is denoted by