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 ${\rho }_2\text{\hspace{0.17em}⁣}:H^{4n}(M;\mathbb{Z})\to H^{4n}(M;\mathbb{Z}\mathrm{/}2)$ denote the $2$ reduction homomorphism. For any torsion class $t\in H^{4n}(M;\mathbb{Z})$, we establish the identity $$⟨{\rho }_2(t)\cdot S{q}^2{\rho }_2(t),[M]⟩=⟨{\rho }_2(t)\cdot S{q}^2{v}_{4n}(M),[M]⟩,$$ where $S{q}^2$ 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 $⟨\cdot ,\cdot ⟩$ denotes the Kronecker product. This result generalizes the work of Landweber and Stong from spin to spin${}^c$ manifolds. As an application, let ${\beta }^{\mathbb{Z}\mathrm{/}2}\text{\hspace{0.17em}⁣}:H^{4n+2}(M;\mathbb{Z}\mathrm{/}2)\to H^{4n+3}(M;\mathbb{Z})$ be the Bockstein homomorphism associated to the short exact sequence of coefficients $\mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}2$. We deduce that ${\beta }^{\mathbb{Z}\mathrm{/}2}(S{q}^2{v}_{4n}(M))=0$, and consequently, $S{q}^3{v}_{4n}(M)=0$, for any closed oriented spin${}^c$ manifold $M$ with $\dim M\le 8n+1$.