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

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.