# Weil homomorphism

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $$G$$ bundle. In this book, they use connection on a principal bundle to define (among other things) the characteristic classes of a (complex) vector bundle.

Let $$G$$ be a Lie group and $$P(M,G)$$ be a principal $$G$$ bundle. Let $$\Gamma$$ be a connection on $$P(M,G)$$. This defines what is called a Weil homomorphism $$I(G)\rightarrow H^*(M,\mathbb{R})$$.

Given a complex vector bundle $$E\rightarrow M$$ with fibre $$\mathbb{C}^r$$ they consider associated principal $$Gl(r,\mathbb{C})$$ bundle $$P\rightarrow M$$ and define $$k$$-th Chern class of $$E$$ to be image of some element of $$I(G)$$.

But, it seems this Weil homomorphism can do more than defining Characteristic classes.

Is Weil homomorphism introduced (and used) only to define Characteritic classes? If not, where else do we use this Weil homomorphism?

• I do not mean to say Weil homomorphism construction is easy or that I understand it completely in full details.. :) :) I see it is very non trivial construction :) :) – Praphulla Koushik Mar 13 at 1:25