Timeline for Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Jul 16, 2013 at 22:11 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
correct minor typos
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| Jul 16, 2013 at 21:02 | answer | added | András Szűcs | timeline score: 2 | |
| Sep 30, 2010 at 15:07 | vote | accept | Dylan Wilson | ||
| Sep 30, 2010 at 15:07 | comment | added | Dylan Wilson | @Scott- Apologies; it felt spammy to post two such highly related questions in the span of a few minutes. | |
| Sep 30, 2010 at 8:24 | comment | added | S. Carnahan♦ | Dylan, if you have a new question, you should submit it as a new question, rather than writing over the old one. | |
| Sep 30, 2010 at 6:20 | answer | added | Christian Nassau | timeline score: 7 | |
| Sep 30, 2010 at 5:22 | comment | added | Dylan Wilson | @Paul- Got it! I'll have to look at the paper then. Looks like this question has been answered. And your assumption was wrong :), I didn't even know what characteristic classes were until about 6 hours ago... so this is all new and fascinating. @Daniel- Those look GREAT! Thank you very much! (I'm looking at the 2001 version; is that the most current?) | |
| Sep 30, 2010 at 3:51 | answer | added | Dev Sinha | timeline score: 3 | |
| Sep 30, 2010 at 3:48 | comment | added | Daniel Pomerleano | THere are some pretty great notes by Haynes Miller called "Notes on Cobordism" that explain all of this and more. | |
| Sep 30, 2010 at 3:27 | comment | added | Paul | I mean Thom determines, not the numbers! For any sequence $(n_1, n_2, n_3,..n_k)$ where $n_1 +2n_2+3n_3+...+kn_k=d$, you get a Stiefel whitney number of a d dimensional manifold by evaluating w_1^n_1 U ... U w_k^n_k on the fundamental class. The collection of {0,1} you get from all such "partions" of d give a monomorphism $Omega_d\to Z/2^p$, i.e. they determine the bordism class for a given manifold. But this need not be onto; eg for d=1, p=1. Thom determined the image. | |
| Sep 30, 2010 at 1:56 | comment | added | Dylan Wilson | Also: Is there any way to fix the typo in my question title without needlessly bumping up my post? | |
| Sep 30, 2010 at 1:55 | comment | added | Dylan Wilson | I'm confused: "Stiefel-whitney numbers... determine which numbers occur." The numbers determine themselves? | |
| Sep 29, 2010 at 23:11 | comment | added | Paul | I'll assume you already know that Thom's original paper both proves that the Stiefel Whitney numbers determine the unoriented bordism class and also determines which numbers occur. Take a look at WIkipedia's "cobordism" page. | |
| Sep 29, 2010 at 23:04 | comment | added | Dylan Wilson | Whoops, I put "classes" when I meant "numbers." I don't know if your comment would give a necessary condition for when some sequence of zeros and ones can be realized as the Stiefel-Whitney numbers of some manifold. Would it? | |
| Sep 29, 2010 at 23:02 | history | edited | Dylan Wilson | CC BY-SA 2.5 |
edited body
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| Sep 29, 2010 at 21:20 | history | edited | Dylan Wilson | CC BY-SA 2.5 |
typo fixed in title
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| Sep 29, 2010 at 21:00 | answer | added | Oscar Randal-Williams | timeline score: 5 | |
| Sep 29, 2010 at 20:56 | history | asked | Dylan Wilson | CC BY-SA 2.5 |