# Wu formula for manifolds with boundary

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!

## 1 Answer

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 $$i:M\hookrightarrow N$$.