Milnor's triple linking number and Haefliger's embedding invariant

In general, the graph $F \colon X \to X \times Y$ of a $f \colong X
\to Y$ sometimes provides important information, e.g.~Lefschetz fixed
point theorem.  In this talk, we will study the ``cobordisms'' of
graphs, or cobordisms of the pairs $(X \times Y, F(X))$ of manifolds
in some sense.  More precisely, for an algebraically split three
component links $L$ in $S^3$, we consider the ``graph''
$\amalg_{i=1}^6 T^3 \to T^3 \times S^3$ associated to $L$.

We generalize Haefliger's invariant of smooth embeddings from $S^3$
into $S^6$ by using the signature of $4$-manifolds,
and then we show that if we apply this invariant to the graph of $L$,
then we obtain Milnor's triple linking number (up to multiplication by
a constant).