Is the "mucube" homeomorphic to the loch ness monster?
I'm wondering if two non-compact 2-dimensional manifolds are homeomorphic. The first is the "loch ness monster" a one ended surface formed from the sum of infinitely many tori:
The other is a surface which I do not know a name for but is related to the "mucube" (a polytope) and so for want of a proper name I am simply calling it that. One way to construct the mucube is to take the tesselation of 3D space by cubes, select all those cubes with at least 2 even coordinates, and then taking the boundary of that set.
Here's a section:
Both of these surfaces are orientable, one-ended, and have infinite genus, so it seems like they might be homeomorphic. I don't have much experience in this type non-compact topology, so I've exhausted all ways I know to tell two surfaces apart. I also can't construct an explicit homeomorphism between the two.
Are they homeomorphic?
The images in this post are my own work. Both released under CCBYSA 4.0. For an SVG version of the loch ness monster image see its page on wikimedia commons
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| User | Comment | Date |
|---|---|---|
| WheatWizard | (no comment) | Jul 4, 2025 at 23:58 |
They are homeomorphic. The conditions given in the question:
- infinite genus
- orientability
- having 1-end
describe exactly one 2-manifold. This is a result of the classification of non-compact surfaces.
The paper by Arredondo and Maluendas "On the infinite Loch ness monster" describes these specific cases the mucube and the Loch Ness monster as being homeomorphic.
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