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We give a simple axiomatic definition of a rational-valued invariant s(W,V,e)of triples (W,V,e), where W is a (smooth, oriented, closed) 6-manifold and V isa 3-submanifold of W, and where e is a second rational cohomology class of thecomplement of V satisfying a certain condition. The definition is stated interms 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 Poincare dual ofa Seifert surface of V, the invariant coincides with -8 times Haefliger'sembedding invariant of (S^6,V). Our definition recovers a more generalinvariant due to Takase, and contains a new definition for Milnor's triplelinking number of algebraically split 3-component links in R^3 that is close tothe one given by the perturbative series expansion of the Chern-Simons theoryof links in R^3.

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