The Fox property for codimension one embeddings of products of three spheres into spheres
Let \(M\) be a smooth closed connected \((n-1)\)-dimensional manifold for which is given a set \(\mathcal S\) of ``standard embeddings'' into the \(n\)-sphere, \(f_\sigma : M \to S^n , \sigma\in \mathcal S\). Denote by \(N_+^\sigma\) and \(N_-^\sigma\) the closures of the two components of \(S^n\setminus f_\sigma(M)\). Two properties of such embeddings are introduced. One is called the ``Alexander property'' and the other one the ``Fox property''. In his paper [``On the subdivision of a 3-space by a polyhedron'', Nat. Acad. Proc. 10, 6-8 (1924; JFM 50.0659.01)], \textit{J. W. Alexander} proved that every torus \(S^1\times S^1 \subset S^3\) bounds a solid torus. As a generalization one says that a smooth embedding \(f:M\to S^n\) has the Alexander property if the closure of one of the two components of \(S^n\setminus f(M)\) is diffeomorphic to \(N_+^\sigma\) or \(N_-^\sigma\) for some \(\sigma\in \mathcal S\). \textit{R. H. Fox} [``On the imbedding of polyhedra in 3-space'', Ann. Math. (2) 49, 462--470 (1948; Zbl 0032.12502)] proved that for a connected closed surface \(F\) in \(S^3\), the closure of any component of \(S^3\setminus F\) is homeomorphic to the closed complement of some regular neighborhood of a finite linear graph. As a generalization one says that a smooth embedding \(f:M\to S^n\) has the Fox property if for some \(\sigma\in \mathcal S\) there exist a pair of smooth embeddings \(f_+:N_+^\sigma \to S^n\) and \(f_-:N_-^\sigma \to S^n\) such that the closures of the two components of \(S^n\setminus f(M)\) are diffeomorphic to the closures of \(S^n\setminus f_{+}(N_+^\sigma)\) and \(S^n\setminus f_-(N_-^\sigma)\). \(n\neq 4\) in the smooth case. The paper treats smooth embeddings \(f:S^1\times S^q\times S^r\to S^q\times S^{r+2}\) and proves that the Alexander property implies the Fox property. Standard embeddings for this problem are defined in the paper. The result says that every smooth embedding of \(f:S^1\times S^q\times S^r\) into \(S^q\times S^{r+2}, 1\leq q\leq r, 3\leq q + r\), has the Fox property, provided \(r \neq q+1\) or \(r=q+1\) is even.
scientific article; zbMATH DE number 5996754
scientific article