# Timeline for Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?

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Jul 16 '13 at 22:11 history edited
correct minor typos
Jul 16 '13 at 21:02 answer timeline score: 2
Sep 30 '10 at 15:07 comment @Scott- Apologies; it felt spammy to post two such highly related questions in the span of a few minutes.
Sep 30 '10 at 8:24 comment Dylan, if you have a new question, you should submit it as a new question, rather than writing over the old one.
Sep 30 '10 at 6:20 answer timeline score: 7
Sep 30 '10 at 5:22 comment @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 '10 at 3:51 answer timeline score: 3
Sep 30 '10 at 3:48 comment THere are some pretty great notes by Haynes Miller called "Notes on Cobordism" that explain all of this and more.
Sep 30 '10 at 3:27 comment 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 '10 at 1:56 comment Also: Is there any way to fix the typo in my question title without needlessly bumping up my post?
Sep 30 '10 at 1:55 comment I'm confused: "Stiefel-whitney numbers... determine which numbers occur." The numbers determine themselves?
Sep 29 '10 at 23:11 comment 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 '10 at 23:04 comment 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 '10 at 23:02 history edited
edited body
Sep 29 '10 at 21:20 history edited
typo fixed in title
Sep 29 '10 at 21:00 answer timeline score: 5
Sep 29 '10 at 20:56 history asked