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Q&A Is the "mucube" homeomorphic to the loch ness monster?

posted 1y ago by WheatWizard‭  ·  edited 1y ago by WheatWizard‭

Answer
#2: Post edited by user avatar WheatWizard‭ · 2025-08-11T12:47:41Z (about 1 year ago)
  • 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.
  • [This paper](https://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc1704/arredo.pdf) describes these specific cases the mucube and the Loch Ness monster as being homeomorphic.
  • 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"](https://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc1704/arredo.pdf) describes these specific cases the mucube and the Loch Ness monster as being homeomorphic.
#1: Initial revision by user avatar WheatWizard‭ · 2025-07-04T20:41:20Z (about 1 year ago) 
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.

[This paper](https://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc1704/arredo.pdf) describes these specific cases the mucube and the Loch Ness monster as being homeomorphic.