A Bilinear Form for Spin${}^c$ Manifolds

Abstract

Let $M$ be a closed oriented spin${}^c$ manifold of dimension $(8n+2)$ with fundamental class $[M]$, and let $\ddot{I}{}_2:H^{4n}(M;\mathbb{Z})\to H^{4n}(M;\mathbb{Z}/2)$ denote the $\mathrm{mod}2$ reduction homomorphism. For any torsion class $t\in H^{4n}(M;\mathbb{Z})$, we establish the identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] 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 $β^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ be the Bockstein homomorphism associated to the short exact sequence of coefficients $\mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\to \mathbb{Z}/2$. We deduce that $β^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $S{q}^3{v}_{4n}(M)=0$, for any closed oriented spin${}^c$ manifold $M$ with $\dim M\le 8n+1$.