A Bilinear Form for Spin$^c$ Manifolds
Huijun Yang
TL;DR
This work extends the Landweber–Stong bilinear form from spin to spin^c manifolds of dimension $(8n{+}2)$ by defining a mod $2$ form on the torsion subgroup $TH^{4n}(M)$ using $\rho_2$ and $Sq^2$ applied to $v_{4n}(M)$. It proves an equivalence: the universal identity for all $x \in H^{4n}(M)$ is equivalent to the same identity for torsion classes $t \in TH^{4n}(M)$, and constructs a universal obstruction class $\Theta \in H^{4n+2}(B\mathrm{Spin}^c; \mathbb{Z}/2)$ encoding this relation. The authors show $\Theta$ is governed by $Sq^1\Theta$, and determine $\Theta$ as $Sq^2 v_{4n}$ for $n \ge 3$ (with a special $n=2$ argument using $\mathbb{H}P^2$); this leads to the main torsion bilinear identity and an application implying $\beta^{\mathbb{Z}/2}(Sq^2 v_{4n}(M))=0$, hence $Sq^3 v_{4n}(M)=0$ in dimensions up to $8n{+}1$. They also discuss the conjectural nonvanishing of $Sq^3 v_{4n}$ for some spin^c $(8n{+}2)$-manifolds, highlighting the nuanced interaction between Wu and Steenrod structures in spin^c geometry.
Abstract
Let $M$ be a closed oriented spin$^{c}$ manifold of dimension $(8n {+} 2)$ with fundamental class $[M]$, and let $ρ_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ denote the $\bmod ~ 2$ reduction homomorphism. For any torsion class $t \in H^{4n}(M;\mathbb{Z})$, we establish the identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] 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 $β^{\mathbb{Z}/2} \colon H^{4n+2}(M; \mathbb{Z}/2) \to H^{4n+3}(M; \mathbb{Z})$ be the Bockstein homomorphism associated to the short exact sequence of coefficients $\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2$. We deduce that $β^{\mathbb{Z}/2}(Sq^2 v_{4n}(M)) = 0$, and consequently, $Sq^3 v_{4n}(M) = 0$, for any closed oriented spin$^{c}$ manifold $M$ with $\dim M \le 8n{+}1$.