On the computation of \(L\)-groups and natural maps
Surgery theory is a basic tool to deal with the classification of compact manifolds. The exact sequence of Sullivan and Wall plays a central role in surgery theory, and it contains Wall's surgery obstruction groups \(L_n(\pi_1)\), where \(\pi_1= \pi_1(M)\) is the fundamental group of the compact smooth (or PL) \(n\)-manifold \(M\) in question. If \(Y\) is another compact smooth (or PL) \(n\)-manifold and \(X\subset Y\) is a closed submanifold of codimension \(q\), one defines the splitting obstruction group \(LS_{n-q}(F)\) which contains the obstruction \(\sigma(f)\) for a simple homotopy equivalence \(f: M\to Y\) to split along \(X\), where \(n-q\geq 5\) and \(F\) is a commutative push-out square of oriented groupoids. For \(q\geq 3\), the splitting obstruction groups coincide with the corresponding Wall surgery obstruction groups. For \(q=1\) and \(X\subset Y\) one-sided, and under some conditions imposed on \(F\), one obtains the Browder-Livesay groups and natural maps, which were computed for many cases. In the paper under review, the authors compute the splitting obstruction groups for many classes of finite (or necessary abelian) 2-groups occurring as the fundamental groups of the manifolds in question.
scientific article; zbMATH DE number 1868563
scientific article