The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H\_n(M;\mathbb{Z}\_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=\sum v\_i\in H^\*(M;\mathbb{Z}\_2)$ is the unique cohomology class such that 
$$\langle v\cup x,z\rangle=\langle Sq(x),z\rangle$$
for all $x\in H^\*(M;\mathbb{Z}\_2)$. Thus, for $k\ge0$, $v\_k\cup x=Sq^k(x)$ for all $x\in H^{n-k}(M;\mathbb{Z}\_2)$, and
$$w\_k(M)=\sum\_{i+j=k}Sq^i(v\_j).$$
Here the Poincare duality guarantees the existence and uniqueness of $v$.


My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class $z\in H\_n(M,\partial M;\mathbb{Z}\_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,\partial M;\mathbb{Z}\_2)\to H\_{n-p}(M;\mathbb{Z}\_2)$$
and
$$D:H^p(M;\mathbb{Z}\_2)\to H\_{n-p}(M,\partial M;\mathbb{Z}\_2).$$


Thank you!



Wu formula for manifolds with boundary
A relative Wu formula for manifolds with boundary is discussed in Section 7 of 


*Kervaire, Michel A.*, [**Relative characteristic classes**](http://dx.doi.org/10.2307/2372561), Am. J. Math. 79, 517-558 (1957). [ZBL0173.51201](https://zbmath.org/?q=an:0173.51201).


In particular, there are *relative Wu classes* $U^q\in H^q(M;\mathbb{Z}/2)$ for $q=0,1,\ldots , n$ defined by the property that 
$$
Sq^q(x)=U^q\cup x \in H^n(M,\partial M;\mathbb{Z}/2)
$$
for all $x\in H^{n-q}(M,\partial M;\mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=M\cup\_{\partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $\partial M$), using naturality with respect to the inclusion $i:M\hookrightarrow N$.


