# Characteristic Classes

# Introduction

Let EXE \ra X be a complex vector bundle. Choose a Hermitian structure on EE and let M=F(E)M = \mcF(E) denote its bundle of unitary frames. So a point of MM consists of a pair (x,e)(x,\bold{e}) where xXx \in X and e=(e1,,en)\bold{e} = (e_{1},\ldots, e_{n}) is an orthonormal basis of ExE_{x}. The group G=U(n)G = \U(n) acts on the right, where

AG:(x,e)(x,eA). A \in G: \quad (x,\bold{e}) \mapsto (x,\bold{e}A).

This makes MM into a principal GG-bundle over XX and hence we get a map

κ:S(g)GH(X)=HG(M). \kappa : S(\mfg^{\ast})^{G} \lra H^{\ast}(X) = H_{G}^{\ast}(M).

The elements of the images of this map gives a subring of the cohomology ring of XX called the ring of characteristic classes of the vector bundle EE. Its definition depended on the choice of Hermitian metric, but it can be shown that κ\kappa does not depend on this choice, justifying the terminology "characteristic".

In the above, we could have taken EE to be a real vector bundle, hh a real scalar product, and MM to be the bundle of orthonormal frames. The group GG would then be the orthonormal group O(n)\O(n). Similarly, we could have taken EE to be an oriented real frame bundle, hh a real scalar product, MM to be the bundle of oriented orthonormal frames, and G=SO(n)G = \SO(n) the special orthogonal group.

In all three cases, U(n),O(n)\U(n), \O(n), and SO(n)\SO(n) we must examine the ring S(g)GS(\mfg^{\ast})^{G} whose structure we shall investigate below. In each case there are standard generators, whose images under κ\kappa are called Chern classes for the case of U(n)\U(n), Pontryagin classes for the case of O(n)\O(n), and one class, called the Pfaffian, in addition to the Pontrygin classes in the case of SO(2n)\SO(2n).

# The Invariants

# G=U(n)G = \U(n)

We can identify the Lie algebra of U(n)\U(n) with the space of all matrices of the form iAiA, where AA is self-adjoint, and hence (using the trace pairing and forgetting the unneeded ii) may identify g\mfg^{\ast} with the space of self-adjoint matrices with the coadjoint action being conjugation:

U:AUAU1,UU(n). U : A \mapsto UAU^{-1},\qquad U \in \U(n).

Define the polynomial cic_{i} of degree ii in AA to be the coefficient of (1)iλni(-1)^{i}\lambda^{n-i} in the characteristic polynomial of AA:

det(λA)=λnc1(A)λn1++(1)ncn(A). \det(\lambda - A) = \lambda^{n} - c_{1}(A)\lambda^{n-1} + \ldots + (-1)^{n}c_{n}(A).

For instance, c1(A)=tr(A)c_{1}(A) = \tr(A) and cn(A)=det(A)c_{n}(A) = \det(A). The polynomials ci(A)c_{i}(A) 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 cic_{i} for a complex vector bundle are called its Chern classes.

# G=O(n)G = \O(n)

We may identify g\mfg and also g\mfg^{\ast} with the space of skew-adjoint matrices. For such a matric we have

det(λA)=det(λAT)=det(λ+A), \det(\lambda - A) = \det(\lambda - A^{T}) = \det(\lambda + A),

so all the coefficient of λni\lambda^{n-i} in the characteristic polynomial of AA vanish when ii is odd. We may write

det(λA)=λn+p1(A)λn2+p2(A)λn4+. \det(\lambda - A) = \lambda^{n} + p_{1}(A)\lambda^{n-2} + p_{2}(A)\lambda^{n-4} + \ldots.

The polynomials pip_{i} of degree p2ip_{2i} generate the ring S(g)GS(\mfg^{\ast})^{G}. The corresponding characteristic classes for a real vector bundle are called its Pontryagin classes.

# G=SO(2n)G = \SO(2n)

The Lie algebra g\mfg is the same for O(2n)\O(2n) so all the pip_{i} are invariant polynomials. There is one additional invariant which is not a polynomial in the pip_{i}, called the Pfaffian. It is defined in the following way: To each AgA \in \mfg and v,wV=R2nv,w \in V = \RR^{2n}, set

ωA(v,w):=(Av,w), \w_{A}(v,w) := (Av,w),

where (,)(-,-) denotes the scalar product. We have

ωA(w,v)=(v,Aw)=(Av,w)=ωA(v,w), \w_{A}(w,v) = (v,Aw) = -(Av,w) = -\w_{A}(v,w),

so ωA2(V)\w_{A} \in \wedge^{2}(V^{\ast}) is an alternating bilinear form and the map AωAA \mapsto \w_{A} is a linear isomorphism. The element

1n!ωAn \frac{1}{n!} \w_{A}^{n}

is an element of 2n(V)\wedge^{2n}(V^{\ast}) and the map A1n!ωAnA \mapsto \tfrac{1}{n!}\w_{A}^{n} depends only on the scalar product and so it O(2n)\O(2n) invariant. However the group SO(2n)\SO(2n) preserves a basis element, vol\vol, of 2n(V)\wedge^{2n}(V^{\ast}), where

vol:=e1e2e2n. \vol := e_{1}^{\ast} \wedge e_{2}^{\ast} \wedge \ldots \wedge e_{2n}^{\ast}.

Here e1,,e2ne_{1},\ldots, e_{2n} is any oriented orthonormgal basis and e1,,e2ne_{1}^{\ast}, \ldots, e_{2n}^{\ast} is the dual basis. We may then define Pfaff(A)\Pfaff(A) by

1n!ωAn=:Pfaff(A)vol. \frac{1}{n!}\w_{A}^{n} =: \Pfaff(A)\cdot \vol.

It is a polynomial function of AA of degree nn which is SO(2n)\SO(2n) invariant. For any AA we can find an oriented orthonormal basis relative to which the matrix AA takes the form

[(0λ1λ10)000(0λ2λ20)000(0λnλn0)]. \begin{bmatrix} \begin{pmatrix} 0 & \lambda_{1} \\ -\lambda_{1} & 0 \end{pmatrix} & 0 & \ldots & 0 \\ 0 & \begin{pmatrix} 0 & \lambda_{2} \\ -\lambda_{2} & 0 \end{pmatrix} & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & \begin{pmatrix} 0 & \lambda_{n} \\ -\lambda_{n} & 0 \end{pmatrix} \end{bmatrix}.

Relative to this orthonormal basis we have

ωA=λ1e1e2++λe2n1e2n \w_{A} = \lambda_{1} e_{1}^{\ast} \wedge e_{2}^{\ast} + \ldots + \lambda e_{2n-1}^{\ast} \wedge e_{2n}^{\ast}


1n!ωAn=λ1λnvol \frac{1}{n!}\w_{A}^{n} = \lambda_{1} \cdot \ldots \cdot \lambda_{n}\cdot \vol

and hence

Pfaff(A)=λ1λn. \Pfaff(A) = \lambda_{1} \cdot \ldots \cdot \lambda_{n}.

On the other hand,

det(A)=λ12λn2, \det(A) = \lambda_{1}^{2} \cdot \ldots \cdot \lambda_{n}^{2},

so we have the general formula

Pfaff2=det. \Pfaff^{2} = \det.

So the square of (2π)nPfaff(2\pi)^{-n}\Pfaff is equal to pnp_{n}. 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, 2n2n-dimensional, oriented vector bundle EE is called the Euler class of the vector bundle, and is denoted by

e(E)H2n(X). e(E) \in H^{2n}(X).