Timeline for Detailed proof of cup product equivalent to intersection
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2010 at 2:08 | comment | added | B. Bischof | Thank you Brian and Kevin, I will take a look at that, it seems to be precisely what I was looking for. Thanks to the other for your suggestions, if Hutchings and the answer below don't work out, I will resort to your suggestions :) | |
| Apr 5, 2010 at 0:04 | comment | added | Tom Church | For de Rham cohomology, I think you can find this in Bott-Tu. (Indeed it's almost immediate from the explicit tubular-neighborhood construction of forms $\omega_M$ representing $\text{PD}(M)$: if $M$ and $N$ are transverse, then $\omega_M\wedge \omega_N$ is equal to $\omega_{M\cap N}$ if you choose your coordinates consistently.) | |
| Apr 4, 2010 at 23:53 | answer | added | Dev Sinha | timeline score: 13 | |
| Apr 4, 2010 at 23:15 | history | edited | Yemon Choi |
changed schubert-varieties tag to schubert-cells as per OP's comment
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| Apr 4, 2010 at 23:05 | comment | added | Kevin H. Lin | Here are the notes that Brian Conrad refers to: math.berkeley.edu/~hutching/teach/215b-2005/cup.pdf | |
| Apr 4, 2010 at 22:59 | comment | added | Tyler Lawson | I believe that Dold is the canonical reference for getting the intersection theory correct. | |
| Apr 4, 2010 at 22:52 | comment | added | BCnrd | Michael Hutchings has some lecture notes on this for cohomology classes of smooth closed submanifolds; key point is to use Thom class. One can also ask for analogue in etale cohomology without smoothness or properness hypotheses, for which one has to modify the technique to make a rigorous proof in such generality. | |
| Apr 4, 2010 at 22:33 | comment | added | Ryan Budney | Alternatively, it's pretty much immediate if you use simplicial homology and the corresponding dual polyhedral decomposition for cohomology. But in that setting "intersection" is a very rigid idea and not quite appropriate for talking about homology of Grassman manifolds. | |
| Apr 4, 2010 at 22:31 | comment | added | Ryan Budney | It's a chapter in Bredon's "Geometry and topology". You can also prove it readily from what's in Hatcher, but I don't think it's assembled that way. You need the Thom isomorphism and its relation to Poincare duality to get anywhere. | |
| Apr 4, 2010 at 22:27 | comment | added | B. Bischof | The tag Schubert-varieties is not quite right, but the tag schubert-cells does not exist. | |
| Apr 4, 2010 at 22:26 | history | asked | B. Bischof | CC BY-SA 2.5 |