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 \̄gmκ from a set of immersions S1 × D2 → X to Z/2Z. The \̄gmκ is defined in connection with frames of tangent bundles. Using the geometric interpretation \̄gmκ of q, we prove that K2(X, U; Z) is a Lagrangian of (K1(∂U;Z),q).