Codimension one embeddings of product of three spheres
Let \(1\leq p\leq q\leq r\) and \(f:S^p\times S^q\times S^r\to S^{p+q+r+1}\) be a smooth embedding. An embedding \(f\) as above is said to be exotic if the closure of neither of the two components of \(S^{p+q+r+1}-\text{Im} f\) is diffeomorphic to a product of two spheres and a disc. The authors studied, in their previous paper [Pac. J. Math. 207, 447-462 (2002; Zbl 1058.57022)], the closure of the components of \(S^{p+q+r+1}-\text{Im} f\) for \(p\geq 2\). In this paper, the authors study the case \(p=1\), and prove, in Theorem 1.1, that there exist mutually distinct embeddings \(f_n:S^1\times S^q\times S^r\to S^{q+r+2}\;(n=1,2,3,\dots )\) such that the closure of neither of the two components of \(S^{q+r+2}-\text{Im} f_n\) is homotopy equivalent to a product of two spheres (and hence the \(f_n\)'s are exotic). Combining this result and those in their paper mentioned above, the authors notice the following: (1) For \(1\leq p\leq q\leq r\), there exists an exotic embedding of \(S^p\times S^q\times S^r\) into \(S^{p+q+r+1}\) if and only if \(p\geq 2, p+q=r\) and \(r\) is odd, or \(p=1\), (2) even though, the closure of one of the two components of \(S^{p+q+r+1}-\text{Im} f\) has the same homology as a product of two spheres and a disc. The authors further study the case \(p=q=r=1\) in detail as follows: For a locally flat topological embedding \(f\) of \(S^1\times S^1\times S^1\) into \(S^4\), the closures of the two components of \(S^4-\text{Im} f\) are homeomorphic to the complements of interiors of an embedded solid torus and Montesinos' twin.
scientific article; zbMATH DE number 2136479