  The paper introduces a group $LSP$ of obstructions for splitting a homotopy
equivalence along a pair of submanifolds. We develop exact sequences relating
the $LSP$-groups with various surgery obstruction groups for manifold triple
and structure sets arising from triples of manifolds. 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.
