Return to Question

It is well-known that Bieberbach proved that any closed flat Riemannian manifold is a quotient of a torus. Let $(M,g)$ be a flat Riemannian $3$-manifold, we know that $(M,g)\cong T^3/\Gamma$, where $\Gamma$ is a finite group.

Q

• Suppose that $M$ is orientable and $b_1(M)=0$ or $H_1(M,\mathbb Z)=0$, can we have a classification for group $\Gamma$ acting on $T^3=\mathbb R^3/\mathbb Z^3$?

• On the other hand: On Wiki(https://en.wikipedia.org/wiki/Flat_manifold), it writes that complete list of the 6 orientable and 4 non-orientable compact examples is related to Seifert fiber space. I think that it means that closed oriented flat(Riemannian) Seifert fiber spaces are classified, could anyone give a hint?

I think above questions were already solved, any reference is welcome!

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.

Questions

• 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!

It is well-known that Bieberbach proved that any closed flat Riemannian manifold is a quotient of a torus. Let $(M,g)$ be a flat Riemannian $3$-manifold, we know that $(M,g)\cong T^3/\Gamma$, where $\Gamma$ is a finite group.

Q

• Suppose that $M$ is orientable and $b_1(M)=0$ or $H_1(M,\mathbb Z)=0$, can we have a classification for group $\Gamma$ acting on $T^3=\mathbb R^3/\mathbb Z^3$?

• On the other hand: On Wiki(https://en.wikipedia.org/wiki/Flat_manifold), it writes that complete list of the 6 orientable and 4 non-orientable compact examples is related to Seifert fiber space. I thins it means that closed oriented flat(Riemannian) Seifert fiber spaces are classified, could anyone give a hint?

I think above questions were already solved, any reference is welcome!

It is well-known that Bieberbach proved that any closed flat Riemannian manifold is a quotient of a torus. Let $(M,g)$ be a flat Riemannian $3$-manifold, we know that $(M,g)\cong T^3/\Gamma$, where $\Gamma$ is a finite group.

Q

• Suppose that $M$ is orientable and $b_1(M)=0$ or $H_1(M,\mathbb Z)=0$, can we have a classification for group $\Gamma$ acting on $T^3=\mathbb R^3/\mathbb Z^3$?

• On the other hand: On Wiki(https://en.wikipedia.org/wiki/Flat_manifold), it writes that complete list of the 6 orientable and 4 non-orientable compact examples is related to Seifert fiber space. I think that it means that closed oriented flat(Riemannian) Seifert fiber spaces are classified, could anyone give a hint?

I think above questions were already solved, any reference is welcome!