A geometric quadratic form of 3-dimensional normal maps

Associated to a homological surgery problem (f, b) consisting of a degree 1 map f : X → Y between compact, oriented, 3-dimensional manifolds and a stable vector bundle map b:T(X) → η covering f, we obtain a butterfly diagram of homology kernels, and a quadratic module (K₁(∂U; ℤ),q), where q:K₁(∂U; ℤ) → ℤ/2ℤ. The quadratic form q is related to a map μ̄_κ from a set of immersions S¹ × D² → X to ℤ/2ℤ. The μ̄_κ is defined in connection with frames of tangent bundles. Using the geometric interpretation μ̄κ of q, we prove that K₂(X, U; ℤ) is a Lagrangian of (K₁(∂U;ℤ),q).