The groups \(LS\) and morphisms of quadratic extensions. (Q1809988)

Let \(U\) be a tubular neighborhood of a submanifold \(X\) of codimension \(q\) in the manifold \(Y\). There is a push-out square \(F\) defined by the diagram of fundamental groups of \(\partial U\), \(X\), \((Y \setminus X)\), and \(Y\). For this geometric situation one has associated Wall surgery obstruction groups \(LS_{n-q}(F)\). This paper describes an algebraic generalization of this that is based on the notion of an anti-structure, which is a triple \((R, \alpha, u)\) where \(R\) is a ring with identity, \(\alpha : R \rightarrow R\) is an antiautomorphism, and \(u \in R^{*}\) is an invertible element, such that \(\alpha(u) = u^{-1}\) and \(\alpha^{2} = uxu^{-1}\) for all \(x \in R\). The notion corresponding to the pushout square is a morphism of quadratic extensions of anti-structures. L-groups are defined for this category that generalize the geometric versions.