Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?
Solution 1:
The answer is no. See https://mathoverflow.net/questions/69085/can-one-use-atiyah-singer-to-prove-that-the-chern-weil-definition-of-chern-classe