TITLE: Vanishing of characteristic numbers vs vanishing of characteristic classes
QUESTION [23 upvotes]: A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same Stiefel-Whitney and Pontryagin numbers (I'll just talk about Stiefel-Whitney for brevity).  An immediate consequence is that if $M$ is a closed manifold which has a non-vanishing Stiefel-Whitney number involving $w_k$ for some $k$, then $w_k(N)\neq 0$ for any $N$ which is bordant to $M$; in other words this non-vanishing characteristic number provides a "bordism reason" for why a characteristic class should be non-zero.
My question concerns the converse.  Given an $M$, suppose that for some $k$ every Steifel-Whitney number involving $w_k$ vanishes, so that there is "no bordism reason" for the class to be non-vanishing: is it then possible to find an $N$ which is bordant to $M$ and has $w_k(N)=0$?  If so, is it possible to simultaneously eliminate all classes which have no bordism reason to be non-zero?
This seems like something which maybe shouldn't be expected since characteristic classes often provide obstructions to doing surgery, and two manifolds are orientedly bordant exactly when they differ by a finite sequence of surgeries.  On the other hand the manifold $\mathbb{CP}^n\#\overline{\mathbb{CP}^n}$ is null-bordant even though it has many non-vanishing characteristic classes for $n>1$ (half the time it isn't even spin).

REPLY [8 votes]: This is not a complete answer, but rather two specific instances where the answer is yes.

If every Stiefel-Whitney number of $M$ involving $w_1$ vanishes, then $M$ is cobordant to an orientable manifold $N$. See Proposition 4 of Wall, C. T. C., Determination of the cobordism ring, Ann. Math. (2) 72, 292-311 (1960). ZBL0097.38801.

If every Stiefel-Whitney number of $M$ involving $w_1$ and $w_2$ vanishes, then $M$ is cobordant to a spin manifold $N$. See Corollary 2.4 of Anderson, D. W.; Brown, Edgar H. jun.; Peterson, Franklin P., The structure of the Spin cobordism ring, Ann. Math. (2) 86, 271-298 (1967). ZBL0156.21605.