Half-turn flat three-manifold
= Half-turn flat three-manifold
This closed <flat Riemannian manifold> is the <mapping torus> of $-I$ on a two-dimensional <flat torus>. Equivalently, take planar lattice translations and the screw $(v,z)\mapsto(-v,z+1/2)$ in Euclidean three-space. Its square is a unit axial translation, its holonomy is $C_2$, and its first <Betti number> is one.