### Abstract

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 (K_{1}(∂U; ℤ),q), where q:K_{1}(∂U; ℤ) → ℤ/2ℤ. The quadratic form q is related to a map μ̄_{κ} from a set of immersions S^{1} × D^{2} → 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 K_{2}(X, U; ℤ) is a Lagrangian of (K_{1}(∂U;ℤ),q).

Original language | English |
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Pages (from-to) | 77-102 |

Number of pages | 26 |

Journal | Topology and its Applications |

Volume | 83 |

Issue number | 2 |

Publication status | Published - Dec 1 1998 |

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

Morimoto, M. (1998). A geometric quadratic form of 3-dimensional normal maps.

*Topology and its Applications*,*83*(2), 77-102.