  Cobordism invariance shows that the index, in K-theory, of a family of
pseudodifferential operators on the boundary of a fibration vanishes if the
symbol family extends to be elliptic across the whole fibration. For Dirac
operators with spectral boundary condition, Dai and Freed \cite{dai-freed1}
gave an explicit version of this at the level of the determinant bundle. Their
result, that the eta invariant of the interior family trivializes the
determinant bundle of the boundary family, is extended here to the wider
context of pseudodifferential families of cusp type.
