Wu formula for manifolds with boundary

Question

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!

Answer 1

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 $U^q\in H^q(M;\mathbb{Z}/2)$ 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 $i:M\hookrightarrow N$.