TY - JOUR

T1 - An invariant of embeddings of 3-manifolds in 6-manifolds and milnor's triple linking number

AU - Moriyama, Tetsuhiro

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2011

Y1 - 2011

N2 - We give a simple axiomatic definition of a rationalvalued invariant σ(W,V,e) of triples (W,V,e), where W ⊃ V are smooth oriented closed manifolds of dimensions 6 and 3, and e is a second rational cohomology class of the complement W \ V satisfying a certain condition. The definition is stated in terms of cobordisms of such triples and the signature of 4-manifolds. When W = S 6 and V is a smoothly embedded 3-sphere, and when e/2 is the Poincaré dual of a Seifert surface of V, the invariant coincides with -8 times Haefliger's embedding invariant of (S 6,V). Our definition recovers a more general invariant due to Takase, and contains a new definition for Milnor's triple linking number of algebraically split 3-component links in ℝ 3 that is close to the one given by the perturbative series expansion of the Chern-Simons theory of links in ℝ 3.

AB - We give a simple axiomatic definition of a rationalvalued invariant σ(W,V,e) of triples (W,V,e), where W ⊃ V are smooth oriented closed manifolds of dimensions 6 and 3, and e is a second rational cohomology class of the complement W \ V satisfying a certain condition. The definition is stated in terms of cobordisms of such triples and the signature of 4-manifolds. When W = S 6 and V is a smoothly embedded 3-sphere, and when e/2 is the Poincaré dual of a Seifert surface of V, the invariant coincides with -8 times Haefliger's embedding invariant of (S 6,V). Our definition recovers a more general invariant due to Takase, and contains a new definition for Milnor's triple linking number of algebraically split 3-component links in ℝ 3 that is close to the one given by the perturbative series expansion of the Chern-Simons theory of links in ℝ 3.

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M3 - Article

AN - SCOPUS:84857537186

VL - 18

SP - 193

EP - 237

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -