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Obstructions to reconstructions from a pair of manifolds

The groups LP* of obstructions to reconstructions from a pair of manifolds X C V have been introduced geometrically by Wall (see [1]). Then an algebraic definition of the groups LP, has been given by Ranicki as for the groups LS, of obstructions to decomposition (see [2]). The groups LS n and LP n can be denned functorially by the universally repelling square F of the oriented fundamental groups and by η mod 4. A pair (Y, X) is a Browder-Livesay pair if X is a one-sided submanifold of codimension 1 and the maps πι(θϋ) —> πι(Υ \ Χ), π\(Χ) —> π\(Υ) are isomorphisms, where U is a tube neighbourhood of X in Y. Then we have the isomorphisms where ν : L n {iti{X) —> L n (iri(dU)) is a transfer. If in this case the condition of isomorphism is weakened to epimorphism, then F is called a geometric diagram (see [3]). Then according to the above there is a close connection between the groups LS*(F) and the L-groups of the groups appearing in the diagram (see [4], [3]). In this case similar connections exist for the groups LP* , as well. homotopic exact sequences of the of the universal squares provide three Levin braids connecting the groups LP*(F) the L-groups.