A geometric quadratic form of 3-dimensional normal maps

Masaharu Morimoto

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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).

Original languageEnglish
Pages (from-to)77-102
Number of pages26
JournalTopology and its Applications
Issue number2
Publication statusPublished - 1998


  • 3-dimensional manifold
  • Lagrangian
  • Quadratic form
  • Surgery

ASJC Scopus subject areas

  • Geometry and Topology


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