TITLE: Stiefel-Whitney Classes over Integers?
QUESTION [21 upvotes]: An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, one can calculate the signature, check that it is non-zero and conclude that it can't be the boundary of an oriented manifold. I decided it might be interesting to calculate the first and only Pontrjagin number to check that it doesn't vanish. I believe Hirzebruch's Signature Theorem can be used to show that it is 6, but I was interested in relating the Stiefel-Whitney classes to the Pontrjagin classes.
I believe one relation is 

$p_i (\mathrm{mod} 2) \equiv w_{2i}^2$  (pg. 181 Milnor-Stasheff)

So I went ahead and did a silly thing. I took my first Chern classes of the original connect sum pieces say 3a and 3b, used the fact that the inclusion should restrict my 2nd second "Stiefel-Whitney Class" (scare quotes because we haven't reduced mod 2) on each piece to these two to get $w_2(connect sum)=(3\bar{a},3\bar{b})$. I can use the intersection form to square this and get $3\bar{a}^2+3\bar{b}^2=6c$ since the top dimensional elements in a connect sum are identified. Evaluating this against the fundamental class gives us exactly the first Pontrjagin number! This is false. Of course this is wrong because it should be 9+9=18 as pointed out below. This does away with my supposed miracle example. My Apologies!
This brings me to a broader question, namely of defining Stiefel-Whitney Classes over the integers. This was hinted at in Ilya Grigoriev's response to Solbap's question when he says 

On thing that confuses me: why are the pullbacks of the integer cohomology of the real Grassmanian never called characteristic classes?

Of course the natural reason to restrict to $\mathbb{Z}/2$ coefficients is to get around orientability concerns. But it seems like if we restrict our orientation to orientable bundles we could use a construction analogous to those of the Chern classes where Milnor-Stasheff inductively declare the top class to be the Euler class, then look at the orthogonal complement bundle to the total space minus its zero section and continue. I suppose the induction might break down because the complex structure is being used, but I don't see where explicitly. If someone could tell me where the complex structure is being used directly, I'd appreciate it. Note the Euler class on odd dimensional fibers will be 2-torsion so this might produce interesting behavior in this proposed S-W class extension.
Another way of extending Stiefel-Whitney classes would be to use Steenrod squares. Bredon does use Steenrod powers with coefficient groups other than $\mathbb{Z}/2$ (generally $\mathbb{Z}/p$ $p\neq 2$), but this creates awkward constraints on the cohomology groups. Is this an obstruction to extending it to $\mathbb{Z}$ coefficients? It would be interesting to see what these two proposed extensions of S-W classes do and how they are related.

REPLY [11 votes]: The integral cohomology rings of both $BO(n)$ and $BSO(n)$ were computed by E. H. Brown, Proceedings AMS, 85, 2, 1982, p. 283-288. These rings are generated by the Pontrjagin classes, Bocksteins of monomials in even Stiefel-Whitney classes and, in the case of $BSO(2k)$, the Euler class. The description is as follows. All torsion is 2-torsion. The subalgebra generated by the Pontrjagin classes (and the Euler class in the case of $BSO(2k)$) has no torsion and is subject to just one relation: the square of the Euler class is the corresponding Pontrjagin class in the $BSO(2k)$ case. The torsion ideal can be identified with the $A$-submodule of the mod 2 cohomology generated by the image of $Sq^1$ where $A$ is the subalgebra generated by the reductions of the Pontrjagin classes, and the reduction of the Euler class in the case of $BS(2k)$. The key observation is Lemma 2.2.
The cohomology of $BO(n)\times BO(m)$ and $BSO(n)\times BSO(m)$ can be described in a similar way. E.H. Brown also computes the images of the Pontrjagin and Euler classes under the Whitney sum maps $BO(n)\times BO(m)\to BO(n+m),BSO(n)\times BSO(m)\to BSO(n+m)$. The Euler classes behave as expected; the torsion component of the images of the Pontrjagin classes is a bit more complicated. Finally, the image of the Bockstein of a monomial in the Siefel-Whitney classes can be computed using Lemma 2.2 and the action of the Steenrod algebra on the mod 2 cohomology.
So ``integral characteristic classes'' do not give any new tools for distinguishing real vector bundles up to isomorphism. However, in principle these classes may give new obstructions to representing bundles as Whitney sums and, by the splitting principle, as tensor products, symmetric or exterior powers etc.