Extension of the Borsuk theorem on non-embeddability of spheres
The authors prove that the suspension \(\sum M\) of a closed \(n\)-manifold \(M\), where \(\geq 1\), does not embed in a product of \(n+1\) curves. In particular, from this result, the following property of spheres discovered by Borsuk, is deduced: the \(n\)-sphere \(\mathbb{S}^{n}\), \(n\geq 2\), does not embed in a product of \(n\) curves. Moreover, the authors generalize the first result for locally connected quasi-\(n\)-manifolds. The proofs are based on the \textit{Second Factorization Theorem} and \textit{Second Structure Theorem} which extend the corresponding results from \textit{A. Koyama, J. Krasinkiewicz} and \textit{S. Spie\(\dot{z}\)} [Trans. Am. Math. Soc. 363, No. 3, 1509--1532 (2011; Zbl 1221.54019). Finally, the authors give a proof of the Second Factorization Theorem for ramified \(3\)-manifolds and they close by asking whether the Factorization Theorem also holds for ramified \(n\)-manifolds.
scientific article; zbMATH DE number 6204275
scientific article