  We give a refined value group for the collection of triple linking numbers of
links in the 3-sphere. Given two links with the same pairwise linking numbers
we show that they have the same refined triple linking number collection if and
only if the links admit homeomorphic surface systems. Moreover these two
conditions hold if and only if the link exteriors are bordant over $B
\mathbb{Z}^n$, and if and only if the third lower central series quotients
$\pi/\pi_3$ of the link groups are isomorphic preserving meridians and
longitudes. We also show that these conditions imply that the link groups have
isomorphic fourth lower central series quotients $\pi/\pi_4$, preserving
meridians.
