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I can't seem to find a reference on the web that gives the $\mathbb{Z}$ cohomology of the Grassmann manifold of real n-planes in infinite dimensional Euclidean space and also the Bockstein maps associated with the coefficient sequence

$$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z/2Z} \to 0.$$

The real question is which products of Stiefel-Whitney classes are really $\mathbb{Z}$ classes.

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See also: mathoverflow.net/questions/16632/… –  Mark Grant Jul 16 '12 at 8:18
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up vote 3 down vote accepted

I don't know if these have everything that you want, but see the following:

Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.

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

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Thanks for the reference I gather from a cursory reading that the products of Stiefel-Whitney class that are mod 2 reductions of integer class are generated by - mod 2 reductions of the Chern classes of the universal n-plane bundle and Sq^1 of the even Stiefel-Whitney classes, that is, the polynomials w1uw2i+ w2i+1. – marc gordon 0 secs ago –  marc gordon Jul 17 '12 at 18:21
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