The paper introduces a group LSP of obstructions to splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the LSP-groups with various surgery obstruction groups for a manifold triple and structure sets arising from a manifold triple. The natural map from the surgery obstruction group of the ambient manifold to the LSP-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some LSP-groups are computed precisely.
