Projective splitting obstruction groups for one-sided submanifolds

A geometric diagram of groups, which consists of groups equipped with geometric antistructures, is a natural generalization of the square of fundamental groups arising in the splitting problem for a one-sided submanifold. In the present paper the groups $LS_*$ and $LP_*$ of such diagrams are defined and the properties of these groups are described. Methods for the computation of $LS_*^p$, $LP_*^p$-groups and natural maps in diagrams of exact sequences are developed in the case of a geometric diagram of finite 2-groups.