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 μ̄κ from a set of immersions S1 × D2 → X to ℤ/2ℤ. The μ̄κ 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).
|Number of pages||26|
|Journal||Topology and its Applications|
|Publication status||Published - Dec 1 1998|
- 3-dimensional manifold
- Quadratic form
ASJC Scopus subject areas
- Geometry and Topology