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 $y\_n \in H^n(M^d; \mathbb{Z})$?



Generalize Wu formula to integral cohomology classes
I do not think such a cohomology operation can exist, as it would descend to a rational cohomology class in $H^{n+4}(K(\mathbb Z,n);\mathbb Q)$. But for any odd $n$ and also for all even $n >4$, this cohomology group is zero! This in turn gives that $P$ is trivial after rationalizing, so the formula you want could never be true.


