### 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 |
---|---|

Pages (from-to) | 77-102 |

Number of pages | 26 |

Journal | Topology and its Applications |

Volume | 83 |

Issue number | 2 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*83*(2), 77-102.

**A geometric quadratic form of 3-dimensional normal maps.** / Morimoto, Masaharu.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 83, no. 2, pp. 77-102.

}

TY - JOUR

T1 - A geometric quadratic form of 3-dimensional normal maps

AU - Morimoto, Masaharu

PY - 1998

Y1 - 1998

N2 - 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).

AB - 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).

KW - 3-dimensional manifold

KW - Lagrangian

KW - Quadratic form

KW - Surgery

UR - http://www.scopus.com/inward/record.url?scp=15944389951&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15944389951&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:15944389951

VL - 83

SP - 77

EP - 102

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 2

ER -