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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?

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I'm not sure how often anyone gets to make a comment like this, but... this question might actually be more suited for MO than math.SE. :-) –  Jesse Madnick Apr 29 '11 at 21:00
    
(Of course others may feel differently. Certainly there are plenty of qualified people here who could answer your question. I just think any question referencing the Atiyah-Singer Index Theorem might be better suited to MO.) –  Jesse Madnick Apr 29 '11 at 21:01
    
I think Jesse is right. An expert's answer to your question could help other non-experts in understanding the Atiyah-Singer theorem, and such an answer would better serve and interest the MO readers than the M.SE ones. Atiyah-Singer is strong voodoo, after all. –  Gunnar Magnusson Apr 29 '11 at 22:24
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@Eric: Certainly there's the algebraic topology proof (or even definition, really) that these are $\mathbb{Z}$-classes, but I saw an analytic argument in one my of lectures the other day that had nothing to do with classifying spaces or normalization. I think it's really cool when the same story can be carried all the way through in two such distinct settings. –  Aaron Mazel-Gee May 2 '11 at 17:14
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@Eric: The case of line bundles is Proposition 4.4.12 in Huybrechts' book, which just chases through the Cech-de Rham complex. I think the proof in my class was different, but maybe it was essentially equivalent. I'll try to find someone who took notes that day... –  Aaron Mazel-Gee May 5 '11 at 9:20
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