On complements of codimension-3 embeddings in \(S^ n\)
This paper explores which groups can be realized as fundamental groups of complements of wildly embedded codimension 3 polyhedra in \(S^ n\), where \(n\geq 7\). The main theorem is the following. Let K be any finite polyhedron, \(1\leq \dim K\leq n-3\), which admits a PL embedding into \(S^ n\), and let \(\pi\) be a countably generated group. Then there exists an (in general non PL) embedding j: \(K\to S^ n\) with \(\pi_ 1(S^ n- j(K))\approx \pi\) if and only if \(\pi\) is perfect and \(H_ 2(\pi)=0\). The proof makes use of techniques from shape theory and decomposition theory.
fundamental groups of complements of wildly embedded codimension 3 polyhedra in \(S^ n\)
scientific article; zbMATH DE number 4094408