## An Invariant of Embeddings of\/ {\Large\bf 3}--Manifolds in\/ {\Large\bf 6}--Manifolds and Milnor's Triple Linking Number

J. Math. Sci. Univ. Tokyo
Vol. 18 (2011), No. 2, Page 193--237.

Moriyama, Tetsuhiro
An Invariant of Embeddings of\/ {\Large\bf 3}--Manifolds in\/ {\Large\bf 6}--Manifolds and Milnor's Triple Linking Number
We give a simple axiomatic definition of a rational--valued invariant $\s(W,V,e)$ of triples $(W,V,e)$, where $W \supset V$ are smooth oriented closed manifolds of dimensions $6$ and $3$, and $e$ is a second rational cohomology class of the complement $W \setminus V$ 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 Poincar\'e 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 $\mathbb{R}^{3}$ that is close to the one given by the perturbative series expansion of the Chern--Simons theory of links in $\mathbb{R}^{3}$.