TITLE: Generalize Wu formula to integral cohomology classes
QUESTION [9 upvotes]: For $\mathbb{Z}_2$ cohomology classes, we have a very useful Wu formula:
In $d$-dimensional manifold and for a $n$-cocycle in $x_n \in H^n(M^d; \mathbb{Z}_2)$, we have $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, where $u_m$ is the $m^{th}$ Wu class, and $Sq^k$ is the Steenrod square.
I wonder if there is a similar formula for integral cohomology classes? For example, if there is a cohomological operation $P^4$, such that
$P^4(y_{d-4}) = p_1 \cup y_{d-4}$ where $p_1$ is the first Pontryagin class and $y_n \in H^n(M^d; \mathbb{Z})$?

REPLY [5 votes]: The Wu classes are, or at least can be, defined by $Sq^{d-n}(x_n) = u_{d-n}\cup x_n$. The interesting part is not that the Wu classes exist, it is that they are related to Stiefel--Whitney classes, which is not at all a priori. Indeed, this implies that SW classes are homotopy-invariant objects (since $Sq^i$ are), which is not at all god-given. Most characteristic classes are not homotopy-invariant, and so will not participate in a Wu-type formula in terms of cohomology operations.