Mryntu

The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $zin H_n(M;mathbbZ_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=sum v_iin H^*(M;mathbbZ_2)$ is the unique cohomology class such that
$$langle vcup x,zrangle=langle Sq(x),zrangle$$
for all $xin H^*(M;mathbbZ_2)$. Thus, for $kge0$, $v_kcup x=Sq^k(x)$ for all $xin H^n-k(M;mathbbZ_2)$, and
$$w_k(M)=sum_i+j=kSq^i(v_j).$$
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 $zin H_n(M,partial M;mathbbZ_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,partial M;mathbbZ_2)to H_n-p(M;mathbbZ_2)$$
and
$$D:H^p(M;mathbbZ_2)to H_n-p(M,partial M;mathbbZ_2).$$

Thank you!

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

Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.

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