= Hantzsche-Wendt manifold
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{wiki=Hantzsche–Wendt_manifold}
The closed orientable <flat Riemannian manifold> with holonomy $C_2\times C_2$ can be presented by Euclidean screws $\alpha(x,y,z)=(x+1/2,-y,-z)$ and $\beta(x,y,z)=(-x,y+1/2,-z+1/2)$. Their squares and the square of their product generate the unit translation lattice. Each nonidentity holonomy coset has a half-integer displacement along its fixed axis, so acts freely. The holonomy has no common fixed vector, giving first <Betti number> zero and a finite <isometry group> for every flat metric.
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