I need to know all relations between Stiefel-Whitney classes for closed manifolds of dimensions 3 and 4. Unfortunately, I found the literature on the subject quite confusing. The answer for all dimensions appears to be contained in E. H. Brown and F. P. Peterson, Bull. AMS 69 (1963), p. 228, but I found it rather cryptic.
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What is cryptic? All relations follow from $u_i=0$ for $2i>\dim X$, where $$ u=\operatorname{Sq}^{-1}w=1+w_1+(w_2+w_1^2)+w_1w_2+(w_4+w_1w_3+w_2^2+w_1^4)+\ldots $$ is the total Wu class. (The reason is the fact that $(\operatorname{Sq}^px)[X]=(u_p\smile x)[X]$ for any class $x\in H^{n-p}(X)$, $n=\dim X$.) Explicitly, in small dimensions we have:
$w_1^2+w_2=w_1w_2=0$ for $\dim X=3$,
$w_1w_2=w_1^4+w_2^2+w_1w_3+w_4=0$ for $\dim X=4$.
Of course, these relations generate an ideal in the algebra of Stiefel--Whitney classes invariant under the Steenrod operations. Hence, in dimension $3$ we also have $w_3=w_1^3=0$, i.e., all classes of dimension $3$ are trivial (any $3$-manifold is null cobordant, Rokhlin's theorem). In dimension $4$, we get, in addition, $w_1w_3=w_1^2w_2=0$.
answered Feb 27, 2014 at 8:06
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$\begingroup$ Why is u_3=w_1 w_2? I thought there is also w_1^3 and w_3 there. $\endgroup$ Commented Feb 27, 2014 at 16:17
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$\begingroup$ Well, that's just the way it is. You need to compute $\def\Sq{\operatorname{Sq}}\Sq^{-1}=1+\Sq^1+\Sq^2+\Sq^2\Sq^1+\ldots$ and use Wu formulas. $\endgroup$ Commented Feb 27, 2014 at 18:16