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

Question

This is just a general curiosity question:

In the standard textbook treatments of characteristic classes, and in particular the treatment of universal Pontrjagin classes, it's standard to consider $H^\ast(BSO,Z[1/2])$ (or $H^\ast(BSO(n),Z[1/2])$) in order to kill the 2-torsion. But I'm curious about that 2-torsion, since it should still give us some extra characteristic class-type information about real oriented bundles. If nothing else, it would give a class that's characteristic in the sense that it behaves the right way with respect to pullbacks (though I imagine it might be too much for these to be stable or have any kind of product formulas). So I suppose my questions are:

1. What's known about such classes?

2. Are they useful for anything?

3. Do these interact in any interesting way with the Stiefel-Whitney classes when everything is reduced mod 2?

4. Why are they usually ignored (or obliterated by coefficient change)?

Answer 1

The basic fact is that the 2-torsion all has order exactly 2, so it injects into the mod 2 cohomology, forming a subalgebra of the polynomial algebra on the Stiefel-Whitney classes. This subalgebra can also be described as the image of the mod 2 Bockstein homomorphism, so it is computable although the answer is not easy to state. The conclusion of all this is that there is no new information in the torsion in the integral cohomology, beyond what one gets from Stiefel-Whitney classes.

These facts are stated as an exercise in Chapter 15 of Milnor and Stasheff's book. A proof can also be found in my notes on Vector Bundles and K-Theory, Theorem 3.16, available on my webpage.

Answer 2

See

Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288

for an answer to your questions. (As for applications: the paper is referenced in other papers and at least one of these uses it to obtain a result in singularity theory.)

If I remember correctly, the computation is quite elaborate, even to state.

Added: the following also does the computations.

Feshbach, Mark, The integral cohomology rings of the classifying spaces of $\text{O}(n)$ and $\text{SO}(n)$. Indiana Univ. Math. J. 32 (1983), no. 4, 511–516.