Cartesian product stabilization of 3-manifolds
It was shown by \textit{K. Borsuk} [Fundam. Math. 33, 273--298 (1945; Zbl 0060.40002)] that any closed manifold has at most one decomposition into the Cartesian product of factors of dimension \(\leq 2\). In the paper under review, the authors investigate the uniqueness of Cartesian product decompositions having a \(3\)-manifold as a factor. Let \(M\) and \(N\) be closed, orientable \(3\)-manifolds which are geometric in the sense of Thurston. Under the natural assumption requiring that \(M\) and \(N\) have no decomposition into Cartesian products, the authors prove that if \(M \times X \approx N \times Y\) (i.e. \(M \times X\) and \(N \times Y\) are homeomorphic) then \(M \approx N\) and \(X \approx Y\) unless \(M\) and \(N\) are Seifert fibered manifolds which are surface bundles over \(S^1\) with the fiber surface of genus \(> 1\), and \(X \approx Y \approx T^2 = S^1 \times S^1\). For Cartesian products with the \(n\)-dimensional torus \(T^n\), they prove that if \(M \times T^n \approx N \times T^n\) for some \(n\), then \(M \approx N\) unless \(M\) and \(N\) are Seifert fibered manifolds as above. The authors also consider uniqueness of Cartesian square decomposition. It was shown by \textit{S. Kwasik} and \textit{R. Schultz} [Topology 41, No.~2, 321--340 (2002; Zbl 0991.57020)] that if two lens spaces have isomorphic fundamental groups, then their Cartesian squares are homeomorphic. On the other hand, in the present paper, the authors prove that if \(M\) and \(N\) are Seifert fibered manifolds which are not lens spaces with the same fundamental group, then \(M \times M\) and \(N \times N\) are homeomorphic only if \(M \approx N\).
scientific article; zbMATH DE number 5791173
scientific article