A bilinear form for spin$^c$ manifolds

Abstract: For any $(8n {+} 2)$-dimensional closed oriented spin$^{c}$ manifold, denote by $[M]$ the fundamental class of $M$, $\langle \cdot ~, ~\cdot \rangle$ the Kronecker product, and $\rho_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ the $\bmod ~ 2$ reduction homomorphism. In this paper, we will prove that the following 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, ] holds for any torsion class $t \in H^{4n}(M;\mathbb{Z})$, where $Sq^2$ is the steenrod square, $v_{4n}(M)$ is the $4n$-th Wu class of $M$ and $x \cdot y$ means the cup product of $x$ and $y$. This is a generalization of the work of Landweber and Stong for spin manifolds. As an application, denote by $\beta^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ the Bockstein homomorphism, we can get that $\beta^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and hence $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$.