Iterated Browder-Livesay invariants and the oozing problem
The Browder-Livesay-Wall surgery theory for codimension 1 submanifolds with nontrivial normal bundle has obstruction groups \(LN_*\). In the case when the fundamental group of the manifold and the submanifold are the same, the fundamental group of the complement is an index 2 subgroup, and the LN-groups are the algebraic l-groups of the corresponding ring with antistructure. This theory can be applied to the detection of the elements in the ordinary surgery obstruction group \(L_*(G)\), which are the closed manifold surgery obstructions. This note investigates the iteration of this procedure for a sequence of such codimension 1 submanifolds, corresponding to a sequence of index 2 subgroups. Theorem 1 states that an element with nontrivial image in one of the relevant LN groups cannot be a closed manifold surgery obstruction. Theorem 2 states that a homotopy equivalence with nontrivial image in one of the relevant LN groups cannot be split along that submanifold.
codimension 1 submanifolds with nontrivial normal bundle
scientific article; zbMATH DE number 4102098