Surgery on triples of manifolds

The surgery obstruction groups for a manifold pair were introduced by Wall for the study of the surgery problem on a manifold with a submanifold. These groups are closely related to the problem of splitting a homotopy equivalence along a submanifold and have been used in many geometric and topological applications. In the present paper the concept of surgery on a triple of manifolds is introduced and algebraic and geometric properties of the corresponding obstruction groups are described. It is then shown that these groups are closely related to the normal invariants and the classical splitting and surgery obstruction groups, respectively, of the manifold in question. In the particular case of one-sided submanifolds relations between the newly introduced groups and the surgery spectral sequence constructed by Hambleton and Kharshiladze are obtained.