Take a solid cube and identify its faces in pairs as shown in Problem 5 of the source sheet. The top and bottom faces and the front and back faces are identified by translations. The left and right faces are identified by a translation composed with a rotation. Compute the fundamental group of the resulting space.
The face-identification diagram appears on page 1 of the source sheet.
Solution
1. First identify the top/bottom and front/back face pairs by the translations shown. The quotient is \(T^2\times I\).
For each fixed left-to-right coordinate, the remaining two coordinate directions have opposite sides identified by translation, producing a torus fiber.
2. The source glyph on the left/right pair differs by a half-turn, so the final quotient is the mapping torus of\[-I:T^2\to T^2.\]
The half-turn reverses both standard circle coordinates of the torus fiber, hence induces \(-I\) on \(\pi_1(T^2)\cong\mathbb Z^2\).
3. Therefore\[\boxed{\pi_1(X)\cong \mathbb Z^2\rtimes_{-I}\mathbb Z}.\]
The fundamental group of a mapping torus of a homeomorphism \(h:Y\to Y\) is the HNN semidirect product \(\pi_1(Y)\rtimes_{h_*}\mathbb Z\).
4. Equivalently,\[\boxed{\pi_1(X)=\langle a,b,t\mid [a,b]=1,\ tat^{-1}=a^{-1},\ tbt^{-1}=b^{-1}\rangle.}\]
Take \(a,b\) for the two torus generators and \(t\) for the mapping-torus direction; 2 gives the two conjugation relations.