Boundaries, eta invariant and the determinant bundle

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.