Timeline for Why is the integral of the second chern class an integer?
Current License: CC BY-SA 2.5
9 events
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| Mar 30, 2011 at 23:24 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Mar 30, 2011 at 15:12 | vote | accept | Greg Graviton | ||
| Mar 29, 2011 at 20:17 | comment | added | Greg Graviton | Thank you, David, I have done as you suggested. | |
| Mar 29, 2011 at 19:34 | comment | added | David E Speyer | I'd be curious to see a direct proof of this, which doesn't go through the construction of $H^k(X, \mathbb{Z})$. Why don't you ask this as a separate question over on math.stackexchange.com ? (It will get closed here, because people here will have no problem assuming that integral cohomology exists.) | |
| Mar 29, 2011 at 19:33 | comment | added | David E Speyer | If you only know deRham cohomology, then you don't know what $H^k(X, \mathbb{Z})$ means! OK, that's a bit unfair. It looks like in practice, you are interested in the subspace of $\omega$ in $H^k(X, \mathbb{R})$ such that, for every $k$-cycle $\sigma$, we have $\int_{\sigma} \omega \in \mathbb{Z}$. This is the image of $H^k(X, \mathbb{Z})$ in $H^k(X, \mathbb{R})$. So it sounds like what you want to know is why wedge product preserves this property of differential forms. | |
| Mar 29, 2011 at 16:44 | comment | added | Greg Graviton | What seems more interesting is the integrality property. Assuming I only know de-Rham cohomology, could you elaborate on how I can detect whether a differential form from $H^k(M,\mathbb{C})$ is already a member of $H^k(M,\mathbb{Z})$? Is there a short reason why the wedge (cup) product of two integral forms is again integral? | |
| Mar 29, 2011 at 16:44 | comment | added | Greg Graviton | D'oh! I mixed up the nomenclature. $F \wedge F$ would be the second Chern character class according to "From Calculus to Cohomology". I'm not familiar with classifying spaces (why is there a map $f : M \to \mathbb{CP}^N$ for any manifold $M$), but I don't care much about that. | |
| Mar 29, 2011 at 11:22 | history | edited | Donu Arapura | CC BY-SA 2.5 |
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| Mar 28, 2011 at 19:00 | history | answered | Donu Arapura | CC BY-SA 2.5 |