Chern-Weil homomorphism and Chern/Pontryagin/Euler class

Question

I am reading Chapter on characteristic classes from Foundations of Differential geometry by Kobayashi and Nomizu. This chapter starts with concept of Chern-Weil homomorphism.

Given a Lie algebra $G$ with $\mathfrak{g}$ as its Lie algebra and a principal $G$ bundle $P(M,G)$ chern Weil homomorphism is a map from $I(G)\rightarrow H^*(M,\mathbb{R})$ where $I(G)$ is the algebra of symmetric multilinear mappings on $\mathfrak{g}$ invariant by $G$ and $H^*(M;\mathbb{R})$ is the deRham cohomology algebra on $M$. This they define fixing a connection and then proves this map is independent of choice of connection. I am able to understand this.

Can some one help me to understand how this Chern-Weil homomorphism is involved in understanding about chern/Pontryagin/Euler classes? Any reference that explains motivation on these characteristic classes is also welcome. I am aware of Milnor’s book.

Answer 1

The first thing to do is to remark that because the Chern-Weil map $w:I(G)\to H^*(M,\mathbb{R})$ is independent of the connection chosen, the image of any invariant polynomial $f\in I(G)$ is a characteristic class of the bundle. Basically, given an isomorphism of principal bundles, the chosen connection on your initial induces a connection on your new bundle; the two so-induced Chern-Weil morphisms agree, so they must agree on $f$. Having said this, you can see that the optimal strategy is just to take generators for the polynomial ring on $\mathfrak{g}$ (i.e., $I(G)$).

Now, the characteristic classes you mention are precisely very good choices of generators for that ring. For example, for complex vector bundles, you consider $\mathrm{Gl}_n(\mathbb{C})$, and note that the $c_i$ in the expression

$\begin{equation} \det (I-\frac{1}{2\pi i}tX)=1+c_1(X)t+....+c_n(X)t^n \end{equation}$

generate the ring of polynomials on $\mathfrak{gl}_n(\mathbb{C})$, and therefore their evaluation at the curvature will generate the image of $I(\mathrm{GL}_n(\mathbb{C})$ through the Weil homomorphism. These are precisely the Chern classes of the bundle.

A good book where all this is explained is Morita's. A good short answer on the motivation of this definition is Henry Horton's answer here; if you have (lot's) of time, you can look up Spivak's A Comprehensive Introduction to Differential Geometry, especial volume 3's Chapter 6 and volume 5's Chapter 13