Knotted contractible 4-manifolds in \(S^4\).
The author studies the possibility of knotting of a contractible polyhedron in a PL manifold in codimension zero and shows by examples that this phenomenon occurs on the level of 4-manifolds. This is realised by a construction of two piecewise linear (or smooth) embeddings, \(e_0\) and \(e_1\) of a certain compact contractible 4-manifold \(M\) into \(S^4\) such that \(\pi_1(S^4\setminus e_1(M))\) is nontrivial and \(\pi_1(S^4\setminus e_0(M))\) is trivial. Since \(e_0\) is homotopic to \(e_1\), the manifold \(M\) knots in \(S^4\). In addition, it is proved that any finitely presented perfect group with a balanced presentation is a knot group of an embedding of some contractible 4-manifold in \(S^4\).
scientific article; zbMATH DE number 2101053