Sbornik: Mathematics 188:3 449–463 c 1997 RAS(DoM) and LMS Matematicheski˘ ı Sbornik 188:3 127–142 UDC 515.1 Groups of obstructions to surgery and splitting for a manifold pair Yu.V. Muranov and D. Repovˇ s Abstract. 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 ques- tion 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 fun- damental group of the manifold. New connections between the groups LP∗ , the Novikov–Wall groups, and the splitting obstruction groups are established. Bibliography: 19 titles. § 1. Introduction Let f : M → Y be a normal map of degree one of smooth (piecewise linear, topological) manifolds of dimension n + q, and let X ⊂ Y be a submanifold of dimension n. Then the groups of obstructions to surgery LP n (F ) for the manifold pair are defined. These groups depend functorially on the push-out square F of fundamental groups with orientation π 1 (∂U ) −→ π 1 (Y \ X )     π 1 (X ) −→ π 1 (Y ), (1.1) where U is a tubular neighbourhood of X and all the maps of fundamental groups are induced by the natural inclusions (see [1] and [2]). The groups LP n (F ) are independent of the category of manifolds (smooth, piecewise linear, topological), as are the Novikov–Wall groups and almost all the natural maps considered in this paper. For this reason we shall use the piecewise linear vocabulary in what follows, pointing out distinctions from other categories when necessary. For n  5 we have the obstruction σ(f,M ) ∈ LP n (F ), which is trivial if and only if there exists a map g such that g is transversal to X , lies in the class of the normal The research of the first author was carried out with the financial support of the Russian Foundation for Fundamental Research (grant no. 96-01-0116a). The research of the second author was carried out with the financial support of the Ministry of Science and Technology of the Republic of Slovenia (grant no. J1-7039-0101-95). AMS 1991 Mathematics Subject Classification. Primary 57R67; Secondary 19J25.