Surgery groups of submanifolds of \(\mathbb{S}^3\)
Let \(\mathcal T\) be a subcomplex of a triangulation \(\mathcal K\) of the \(3\)-sphere \(S^3\), and let \(N(\mathcal T)\) be a regular neighbourhood of \(\mathcal T\). Let \(M = \overline{S^3 \smallsetminus N(|\mathcal T|)}\) be the subcomplex complement of \(\mathcal T\), and let \(C\) be a component of \(M\). The main result of the paper shows that the homotopy-topological structure set \(\mathcal S(C\times D^n, \partial (C \times D^n))\) is trivial for \(n \geq 2\) provided that \(M\) is irreducible and \(\partial C\) is incompressible, i.e., the map \(\pi_1(\partial C) \rightarrow \pi_1(C)\) is injective. The proof uses an extension of the Topological Rigidity Theorem of \textit{F. T. Farrell} and \textit{L. E. Jones} [Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)]. As a consequence the author computes the surgery obstruction groups \(L^s_n(\pi_1(C))=L^h_n(\pi_1(C))\).
scientific article; zbMATH DE number 1395066