Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 Þór Magnússon Apr 29 '11 at 22:24
@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
@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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.