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

TL;DR

This paper introduces a rational-valued invariant for triples involving 6-manifolds and 3-submanifolds, unifying and extending previous invariants, including Milnor's triple linking number, through a cobordism-based axiomatic approach.

Contribution

It provides a new axiomatic definition of an invariant that generalizes Milnor's triple linking number and relates to Chern-Simons theory, applicable to embeddings in 6-manifolds.

Findings

Invariant coincides with Haefliger's embedding invariant in specific cases

Recovers a more general invariant by Takase

Provides a new perspective on Milnor's triple linking number

Abstract

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 is a 3-submanifold of W, and where e is a second rational cohomology class of the complement of 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 Poincare 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.