A Bilinear Form for Spinᶜ Manifolds

Let M be a closed oriented spinᶜ manifold of dimension (8n + 2) with fundamental class $[M]$, and let ρ₂ H⁴ⁿ(M; Z) → H⁴ⁿ(M; Z/2) denote the ~ 2 reduction homomorphism. For any torsion class t ∈ H⁴ⁿ(M;Z), we establish the identity \[ ρ_2(t) · Sq^2 ρ_2 (t), [M] = ρ_2 (t) · Sq^2 v₄ₙ(M), [M], \] where Sq² is the Steenrod square, v₄ₙ(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 βZ/2 H⁴ⁿ⁺²(M; Z/2) → H⁴ⁿ⁺³(M; Z) be the Bockstein homomorphism associated to the short exact sequence of coefficients Z × 2 Z → Z/2. We deduce that βZ/2(Sq² v₄ₙ(M)) = 0, and consequently, Sq³ v₄ₙ(M) = 0, for any closed oriented spinᶜ manifold M with M ≤ 8n+1.