= Free dihedral action on a drilled torus
{title2=$|D_n|=2n,\quad g=2n+1$}
Let $D_n$ act on <Euclidean space> by rotations through $2\pi/n$ about a vertical axis and half-turns about $n$ equally spaced horizontal axes. A round <solid torus> about the vertical axis, drilled through radially along the $2n$ places where the horizontal axes cross its tube, has a smooth <boundary> of <genus> $2n+1$ after equivariant rounding of its rims. All nonidentity rotation axes lie in the complement of this <boundary>, so the induced <group action> is a <free action of a group>. Each bore replaces two <closed discs> by an <annulus>, lowering <Euler characteristic> by two and increasing <genus> by one. The <boundary orientation> is preserved; its induced <Riemann surface> has an unramified <quotient space> of <genus> two by the <Riemann-Hurwitz formula>.
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