There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are determined by their holonomy groups
$$
C_1,C_2,C_3,C_4,C_6
$$
and
$$
C_2 \times C_2
$$
The five with cyclic holonomy all arise as the mapping torus of a mapping class of $ T^2 $ with the corresponding order: 1,2,3,4, or 6. These five Euclidean manifolds with cyclic holonomy can even be constructed as a quotient of the special Euclidean group $ SE_2 $ by a cocompact lattice constructed as the semidirect product of a lattice in $ \mathbb{R}^2 $ and a finite cyclic subgroup of $ SL_2(\mathbb{Z}) $ preserving that lattice. For example $ C_1 $ corresponds to the three torus $ T^3 $.
I am very curious about the compact flat orientable 3 manifold with holonomy $ C_2 \times C_2 $ (known as the Hantzsche-Wendt manifold). It is not a mapping torus of $ T^2 $ like the other five, but perhaps it is a mapping torus of the Klein bottle $ K $?
Best Answer
The remaining orientable manifold is called the Hantzsche-Wendt manifold $M^{HW}$, and is not a mapping torus over the Klein bottle. It has first homology $H_1(M^{HW}, \mathbb{Z}) = \mathbb{Z}_4 \times \mathbb{Z}_4$. Any mapping torus $MT(M, f)$ of a manifold $M$ via the map $f: M \to M$ has $\pi_1(MT(M, f)) \cong \pi_1(M_0) \rtimes_{f_*} \mathbb{Z}$. Abelianizing and using Hurewicz's theorem yields that if $M$ is a mapping torus of the Klein bottle, $H_1(M, \mathbb{Z})$ must be of the form $\mathbb{Z} \times H$, where $H$ is a quotient of $H_1(K, \mathbb{Z}) = \mathbb{Z} \times \mathbb{Z}_2$, and hence cannot be $\mathbb{Z}_4 \times \mathbb{Z}_4$. It is worth mentioning that the $3$-torus is a normal covering space of all flat compact $3$-manifolds.
The Hantzsche-Wendt Manifold has a nice description as a branched cover of the complement of the Borromean rings, which is described in Zimmerman's paper On the Hantzsche-Wendt Manifold. The standard reference for a classification of compact flat 3-manifolds is J. Wolf's book Spaces of Constant Curvature, which gives a classification of them and explicit constructions as quotients of $\mathbb{R}^3$ by isometries.
A resource to develop visual intuition about these manifolds is Jeffrey Weeks' program Curved Spaces, which simulates how it would look to "fly around inside of" manifolds and has a number of flat 3 manifolds as pre-built examples, including the Hantzsche-Wendt manifold.