Similarity and diffeomorphism classification of \(S^2\times{\mathbb R}\) manifolds (Q2733944)

\textit{W. P. Thurston} proved in [Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982; Zbl 0496.57005)] that any \(3\)-dimensional homogeneous geometry admitting a compact quotient is equivalent to either \(E^3\), \(S^3\), \(H^3\), \(S^2\times{\mathbb R}\), \(H^2\times{\mathbb R}\), \(\widetilde{\text{SL}_2{\mathbb R}}\), \(\mathbf{Nil}\) or \(\mathbf{Sol}\). The classification problem for compact quotients of these spaces, the so-called space forms, has been solved for \(S^3\) and \(E^3\) (with obvious applications to crystallography), but remained open for the other geometries. NEWLINENEWLINENEWLINERecently, \textit{J. Z. Farkas} [Beitr. Algebra Geom. 42, No. 1, 235-250 (2001; Zbl 0983.20046)] listed all isometry subgroups~\(G\) of \(S^2\times {\mathbb R}\) which give rise to compact space forms \((S^2\times {\mathbb R})/G\), up to similarity equivalence. He found infinite series of non-similar space-forms with local \(S^2\times{\mathbb R}\) metric in this way. In the present paper, the authors consider diffeomorphism equivalence for these space forms and find only four different classes: two orientable and two non-orientable ones. These four classes have appeared in the literature before [\textit{P. Scott,} Bull. Lond. Math. Soc. 15, 401-487 (1983; Zbl 0561.57001) and \textit{J. L. Tollefson,} Proc. Am. Math. Soc. 45, 461-462 (1974; Zbl 0294.57006)], but the presentation here is more complete.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].