A Bilinear Form for Spin$^c$ Manifolds

Let $M$ be a closed oriented spin$^{c}$ manifold of dimension $(8n {+} 2)$ with fundamental class $[M]$, and let $\rho_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ denote the $\bmod ~ 2$ reduction homomorphism. For any torsion class $t \in H^{4n}(M;\mathbb{Z})$, we establish the identity $$\langle \rho_2(t) \cdot Sq^2 \rho_2 (t), [M] \rangle = \langle \rho_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle,$$ where $Sq^2$ is the Steenrod square, $v_{4n}(M)$ is the $4n$-th Wu class of $M$, $x\cdot y$ denotes the cup product of $x$ and $y$, and $\langle \cdot ~, ~\cdot \rangle$ denotes the Kronecker product. This result generalizes the work of Landweber and Stong from spin to spin$^c$ manifolds. As an application, let $\beta^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ be the Bockstein homomorphism associated to the short exact sequence of coefficients $\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2$. We deduce that $\beta^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$.