Groups of obstructions to surgery and splitting for a manifold pair

The surgery obstruction groups $LP_*$ of manifold pairs are studied. An algebraic version of these groups for squares of antistructures of a special form equipped with decorations is considered. The squares of antistructures in question are natural generalizations of squares of fundamental groups that occur in the splitting problem for a one-sided submanifold of codimension 1 in the case when the fundamental group of the submanifold is mapped epimorphically onto the fundamental group of the manifold. New connections between the groups $LP_*$, the Novikov–Wall groups, and the splitting obstruction groups are established.