EDIT: The original question was answered very quickly (and very nicely!) but the answer leads to a pretty obvious subsequent question, which I will now ask. The original question is maintained for motivational purposes below.

I now know that not every sequence of zeros and ones can be realized as the Stiefel-Whitney numbers for some manifold- as I'm sure many of you all already knew. What I don't know, and what I suspect is a more delicate question, is: Which ones are? Is there a relatively easy necessary condition? Any sufficient conditions?

Along similar lines: are there estimates as to the number of cobordism classes in any dimension that are tighter than the number of "possible" Stiefel-Whitney numbers? A tighter bound, as it were.

Thanks!

(original question)

Well I just learned a very cool fact over tea: apparently there are finitely many (unoriented) cobordism classes of compact manifolds in any given dimension! The cobordism class is completely determined by the Stiefel-Whitney characteristic numbers (which were explained to me as "the various numbers one gets by cupping characteristic classes of the tangent bundle together and applying them to the fundamental class, all mod 2")... so that's pretty awesome.

While I get over this initial shock, I was wondering if anyone knew the answer to the following: we have an upper bound on the number of cobordism classes by looking at the number of possible Stiefel-Whitney numbers. But is this upper bound realized?

In other words, given a sequence of zeros and ones (the right number of them), can I always construct a manifold that has precisely that sequence of zeros and ones as its Stiefel-Whitney numbers?

alsodetermines which numbers occur. Take a look at WIkipedia's "cobordism" page. $\endgroup$ – Paul Sep 29 '10 at 23:11Thomdetermines, 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. $\endgroup$ – Paul Sep 30 '10 at 3:27