A Bilinear Form for Spinᶜ Manifolds

Let M be a closed oriented spin^c manifold of dimension (8n + 2) with fundamental class M, and let ρ₂ H^4n (M; Z) H^4n (M; Z/2) denote the ~ 2 reduction homomorphism. For any torsion class t H^4n (M;Z), we establish the identity \ ρ₂ (t) Sq² ρ₂ (t), [M = ρ₂ (t) Sq² v₄₍ (M), M, \] where Sq² is the Steenrod square, v₄₍ (M) is the 4n-th Wu class of M, x y denotes the cup product of x and y, and ~, ~ denotes the Kronecker product. This result generalizes the work of Landweber and Stong from spin to spinᶜ manifolds. As an application, let β^Z/2 H^4n+2 (M; Z/2) H^4n+3 (M; Z) be the Bockstein homomorphism associated to the short exact sequence of coefficients Z 2 Z Z/2. We deduce that β^Z/2 (Sq² v₄₍ (M) ) = 0, and consequently, Sq³ v₄₍ (M) = 0, for any closed oriented spin^c manifold M with M 8n+1.