Stiefel-Whitney Classes over Integers?

I'm grateful to Allen Hatcher, who pointed out that this answer was incorrect. My apologies to readers and upvoters. I thought it more helpful to correct it than delete outright, but read critically.

If $X$ and $Y$ are cell complexes, finite in each degree, and two maps $f_0$ and $f_1\colon X\to Y$ induce the same map on cohomology with coefficients in $\mathbb{Q}$ and in $\mathbb{Z}/(p^l)$ for all primes $p$ and natural numbers $l$, then they induce the same map on cohomology with $\mathbb{Z}$ coefficients. To see this, write $H^n(Y;\mathbb{Z})$ as a direct sum of $\mathbb{Z}^{r}$ and various primary summands $\mathbb{Z}/(p^k)$, and note that the summand $\mathbb{Z}/(p^k)$ restricts injectively to the mod $p^l$ cohomology when $l\geq k$. One can take only those $p^l$ such that there is $p^l$-torsion in $H^\ast(Y;\mathbb{Z})$. (I previously claimed that one could take $l=1$, which on reflection is pretty implausible, and is indeed wrong.)

We can try to apply this to $Y=BG$, for $G$ a compact Lie group. For example, $H^{\ast}(BU(n))$ is torsion-free (and Chern classes generate the integer cohomology), and so rational characteristic classes suffice. In $H^{\ast}(BO(n))$ and $H^{\ast}(BSO(n))$ there's only 2-primary torsion. That leaves the possibility that the mod 4 cohomology contains sharper information than the mod 2 cohomology. It does not, because, as Allen Hatcher has pointed out in this recent answer, all the torsion is actually 2-torsion.

It's sometimes worthwhile to consider the integral Stiefel-Whitney classes $W_{i+1}=\beta_2(w_i)\in H^{i+1}(X;\mathbb{Z})$, the Bockstein images of the usual ones. These classes are 2-torsion, and measure the obstruction to lifting $w_i$ to an integer class. For instance, an oriented vector bundle has a $\mathrm{Spin}^c$-structure iff $W_3=0$.

[I'm sceptical of your example in $2\mathbb{CP}^2$. So far as I can see, $3a+3b$ squares to 18, not 6, and indeed, $p_1$ is not a square.]

Of course any cohomology class of $BG$, with any coefficients, serves as a characteristic class of $G$-principal bundles ($G$ is an arbitrary group). This is more or less the definition of characteristic classes. However, if $G=O(n)$ or $SO(n)$, it is quite difficult to get a hand on integral coefficients. John Klein gave a link here:

What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?

To see the essential ingredients for the definition of Stiefel-Whitney classes for real vector bundles (and similar series), it is helpful to ignore Milnor-Stasheff and forget about the cell decompositions of Grassmannians for a moment. (I learnt the following definition from Matthias Kreck) Let $V \to X$ be a real $n$-dimensional vector bundle and $L \to RP^{\infty}$ be the universal line bundle. The external tensor product $V \boxtimes L$ is a bundle over $X \times RP^{\infty}$. It has an Euler class $e \in H^n (X \times RP^{\infty};Z/2)$. Use Kuenneth to write this group as $\oplus_{k=0,...,n} H^k (X) \otimes H^{n-k} (RP^{\infty})$. Under this isomorphism, $e$ becomes $\sum_k w_k(V) \otimes x^{n-k}$, $x$ the generator of $H^{\ast}(RP^{\infty})$.

The same construction yields the Chern classes, replacing $Z/2$ by $Z$ and $R$ by $C$ throughout.

What you see from this construction is that if you wish to have integral classes, you need the Euler class, i.e. orientability. But, no matter whether $V$ is oriented or not, the bundle $V \boxtimes L$ is not oriented.

What you can do is to replace $L \to RP^{\infty}$ by the universal $2$-dimensional oriented vector bundle $U \to BSO(2)=CP^{\infty}$. The point is that $U$ and hence $V \boxtimes U$ is a complex vector bundle and hence oriented. More precisely

$$V \boxtimes_R U \cong V \boxtimes_R (C \otimes_C U) = (V \otimes_R C) \boxtimes_C U.$$

You get the Pontrjagin classes! You can play the same game with the quaternions and the universal quaternionic line bundle $H \to HP^{\infty}$. Here it is important that for each quaternionic vector bundle $V \to X$, the bundle $V \boxtimes_H H$ is only real oriented and not complex. The classes obtained in this way are also called Pontrjagin classes.

Having defined these classes, one computes the cohomology of the classifying spaces $BG(n)$ ($G=U,O,SO,Sp$) with different coefficient rings $A$ with the help of the Gysin sequence of the sphere bundle $BG(n) \to BG(n+1)$ and induction on $n$. An important point is that the computation goes smoothly if (and only if!) the Euler numbers of the occuring spheres are either zero or invertible (in $A$). Of course, the two cases produce quite different looking results.

If $G=U$ or $G=Sp$, all spheres are odd-dimensional and thus have zero Euler number. Thus the compuation goes well for any $A$.

If $G=O,SO$, then there are even-dimensional spheres around, with Euler number $2$. Therefore the computation is smooth with $Z/2$-coefficients and also if $2$ is invertible in the coefficient ring. But the results are really different in characteristic $2$ and $\neq 2$! If $2$ is neither zero nor invertible in the coefficient ring, things become messy at this point.

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.