Abstract
In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups $LS_*$ and the surgery obstruction groups $LP_*$ for a manifold pair. Finally, we introduce the lower $LS_*$- and $LP_*$-groups, and describe connections between them and the corresponding lower $L_*$-groups and surgery exact sequence.
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