Show that octagon pasting is homeomorphic to the \(T = \mathbb{R}^2 / \mathbb{Z}^2\).
Solution
1. The retained statement is under-specified: no octagon edge-pairing is given.
The original archived Justin's Problems.pdf contains
exactly the sentence in this card and no accompanying octagon diagram;
the legacy rendered solutions page likewise has an empty solution entry
and no image reference. An octagon by itself does not determine a
quotient space: the equivalence relation on its boundary is essential
data.
2. Different edge-pairings of an octagon give non-homeomorphic spaces, so the missing data cannot be reconstructed from the word “octagon”.
For example, pairing the four pairs of opposite sides of a regular octagon by translations gives the standard closed orientable surface of genus \(2\), whose first homology is \(\mathbb Z^4\), so it is not a torus. On the other hand, subdivide each side of the standard square fundamental polygon for the torus into two segments; after a small deformation the resulting fundamental polygon is an octagon, and the induced pairings give a torus. Thus at least two inequivalent “octagon pastings” exist.
3. For the intended torus-type octagon pasting—namely, any octagon obtained by subdividing/deforming the square fundamental polygon while retaining the top-bottom and left-right translation identifications—the quotient is homeomorphic to\[\boxed{\mathbb R^2/\mathbb Z^2.}\]
Let \(Q=[0,1]^2\) with \((0,y)\sim(1,y)\) and \((x,0)\sim(x,1)\). Subdividing boundary edges and applying a homeomorphism from \(Q\) onto an octagonal disc does not change the quotient relation: conjugating the square identifications by that homeomorphism gives the octagon pasting. Therefore the octagon quotient is homeomorphic to \[Q/\sim\;\cong\mathbb R^2/\mathbb Z^2,\] the latter homeomorphism being induced by \((x,y)\mapsto[(x,y)]\) modulo integer translations.
4. Hence a rigorous proof of the unconditional printed sentence is impossible until the missing edge-pairing diagram or equivalent boundary-identification data is restored.
By 2, the conclusion depends on that omitted datum. Step 3 proves the result once the intended torus-type pairing is specified.