By a theorem of L. Bieberbach we know that that every closed flat Riemannian manifold is a quotient of a torus via action of a finite group $\Gamma$. In this question we are interested in the particular case of $3$ dimensional flat riemannian manifolds. So we assume that $(M,g)$ is a flat Riemannian $3$-manifold. then by above theorem we know that $(M,g)\cong T^3/\Gamma$, where $\Gamma$ is a finite group.


  • Is there a complete classification of all finite group actions on flat $3$-torus which resulting quotion space would be a an orientable manifold with vanishing first Betti number or first integral homology?

  • According to Wikipedia Wiki(https://en.wikipedia.org/wiki/Flat_manifold), there is a complete list of all 6 orientable and 4 non-orientable flat compact manifolds. This list consists of all Seifert fiber spaces. Does the link of wikipedia actually means that all orientable Seifert Hyper spaces of dimension $4$ and $6$ have been classified? Any reference is welcome!

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    $\begingroup$ A classification can be found in "Spaces of Constant Curvature" by Joseph Wolf. In particular, there is a unique orientable flat 3-manifold with zero first Betti number, the so called Hantzsche–Wendt manifold. $\endgroup$ Aug 15, 2019 at 11:54
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    $\begingroup$ @IgorBelegradek Thanks a lot. Do you know how it is related to Seifert manifold? $\endgroup$
    – DLIN
    Aug 15, 2019 at 12:02
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    $\begingroup$ Seifert manifold structures on flat 3-manifolds are described in P.Scott's "The geometries of 3-manifolds", see pp/443-448 in homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf. $\endgroup$ Aug 15, 2019 at 12:16
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    $\begingroup$ For an exposition of these manifolds, check out this paper by Colin Adams and Joey Shapiro: The Shape of the Universe: Ten Possibilities - American Scientist americanscientist.org/sites/...org/files/200522415348_306.pdf $\endgroup$
    – Ian Agol
    Aug 15, 2019 at 21:34
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    $\begingroup$ @YCor: for euclidean 3-manifolds homemorphism, diffeomorphism and affine-linear homeomorphism are all equivalent. As you say, the isometry question is significantly more subtle as these manifolds have volumes and shortest closed geodesics, etc. $\endgroup$ Aug 16, 2019 at 6:58


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