Cartesian powers of 3-manifolds
In a previous paper [Topology 41, 321--340 (2002; Zbl 0991.57020)], the authors showed that two 3-dimensional lens spaces \(L\) and \(L^{\prime }\) with isomorphic fundamental groups have diffeomorphic Cartesian squares; i.e., one has \( L\times L\sim L^{\prime }\times L^{\prime }.\) It should be noted that \(L\) and \(L^{\prime }\) do not even have to be homotopy equivalent. The question on the existence of spaces with this property was originally formulated by \textit{S. Ulam} in 1933, and as indicated by the authors [op. cit.], lens spaces are the lowest dimensional compact examples that one can produce. It is natural to ask what other sorts of 3-dimensional examples of this sort exist. Examples of noncompact (open) 3-manifolds \(W^{3}\) such that \(W^{3}\nsim \mathbb{R}^{3},\) but \(W^{3}\times W^{3}=\mathbb{R}^{6}\) have been known for some time [cf. \textit{J. Glimm}, Bull. Soc. Math. Fr. 88, 131--135 (1960; Zbl 0094.36101); \textit{K. W. Kwun}, Ann. Math. 79, 104--108 (1964; Zbl 0119.38602) or \textit{D. R. McMillan Jr.}, Bull. Am. Math. Soc. 67, 510--514 (1961; Zbl 0116.40802)]; one such example is the Whitehead 3-manifold that is contractible but not simply connected at infinity [\textit{J. H. C. Whitehead}, Quart. J. Math. 6, 268--279 (1935; JFM 61.0607.01)]. These suggest that one should concentrate on the existence of compact 3-manifolds \(A,B\) such that \(A\times A\) is homeomorphic to \( B\times B\) but \(A\) is not homeomorphic to \(B.\) The purpose of the present paper is to show that no examples of this sort exist in a large class of basic 3-manifolds whose fundamental groups are either infinite or finite but noncyclic. A topological pair \(X,Y\) is called exponentially stable if \( \prod^{n}\) \(X\approx \prod^{n}\) \(Y\) (here \(\approx \) stands for homeomorphic). The authors prove the following main results. Theorem 2. Let \(M\) and \(N\) be lens spaces with isomorphic fundamental groups. Then the pair \(M,\) \(N\) is NOT exponentially stable if and only if \(M\) and \(N\) are homotopy equivalent but nonhomeomorphic lens spaces. Theorem 3. Let \(M\) and \(N\) be closed, connected, oriented, and connected sums of geometric 3-manifolds such that \(H^1(M;\mathbb{Z})=H^1(N;\mathbb{Z})=0\) and one of \(M,N\) has no lens spaces in its prime decomposition. Then \(\prod^{n} M\approx \prod^{n} N\) for some \(n\geq 2\) if and only if \(M\approx N\) (here the term of geometric manifold is in the sense of \textit{W. Thurston} [Bull. Am. Math. Soc., New Ser. 6, 357--379 (1982; Zbl 0496.57005)]). In particular, the authors deduce that (1) two closed, connected, oriented, irreductible, geometric 3-manifolds with vanishing first Betti number are exponentially stable if and only if they are not both lens spaces with isomorphic fundamental groups, (2) two irreducible Seifert manifolds with vanishing first Betti number are exponentially stable if and only if they are not both lens spaces with isomorphic fundamental groups, and (3) two manifolds satisfying the condition of Theorem 3 are exponentially stable.
scientific article; zbMATH DE number 5080163
scientific article