Surface systems and triple linking numbers

We characterise when two links in the 3–sphere admit homeomorphic surface systems, where a surface system is a collection of embedded surfaces with boundary the link. The answer is in terms of 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 BZ n, and if and only if the third lower central series quotients π/π3 of the link groups are isomorphic preserving meridians and longitudes