Knotted homology 3-spheres in \(S^ 5\)
The isotopy class of a smoothly embedded homology 3-sphere K in \(S^ 5\) is called a simple 3-knot if \(\pi _ 1(S^ 5-K)\cong {\mathbb{Z}}\). Simple 3-knots are classified by their Seifert matrices. Using this classification the author defines a trivial \(\Sigma\)-knot for every homology 3-sphere \(\Sigma\) with zero Rohlin invariant and characterizes it as the unique simple \(\Sigma\)-knot whose complement is homotopy equivalent to \(S^ 1\). Furthermore the following is proved: (1) There exist infinitely many decomposable algebraic 3-knots. (2) There exist a simple fibered 3-knot which is, though decomposable, not the connected sum of two nontrivial fibered 3-knots. (3) Algebraically fibered simple 3-knots are stably fibered.
smoothly embedded homology 3-sphere in \(S^ 5\)
scientific article; zbMATH DE number 4003075