Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes? In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F_\nabla$, and then we get Chern classes of $E$ from the curvature form.  A priori it looks like these live in $H^*(X;\mathbb{C})$, but by an argument that I don't feel like I really understand, they're in the image of $H^*(X;\mathbb{Z})$, which is where they're usually considered to actually live.  I've also recently been learning about the Atiyah-Singer index theorem, and I get the impression that whenever I see a arbitrary constants in geometry that end up having to live in $\mathbb{Z}$ I should ask myself whether the index theorem is lurking in there somewhere.  Is there anything to this wild guess?

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
