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Can all different topologies of manifolds be encoded (isomorphically?) as graphs? How should I formulate this question in google to quickly find answer?

DSblizzard
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  • For topological manifolds it is a difficult issue. But if you have a triangulated manifold, then yes. The graph in question will be the 1-skeleton of the triangulation with those cliques marked which correspond to top-dimensional simplices. – Moishe Kohan Oct 02 '23 at 15:10
  • @Moishe Kohan: Can you please give a link to paper which discusses this issue in general case or at least mentions it? – DSblizzard Oct 02 '23 at 15:17
  • Do you need a reference to the notion of a triangulated manifold? From that, you would have to work things out yourself, but it is very straightforward. – Moishe Kohan Oct 02 '23 at 15:20
  • No, I want to know about what you wrote "For topological manifolds it is a difficult issue". – DSblizzard Oct 02 '23 at 15:22
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    Read Manolescu's paper https://arxiv.org/abs/1607.08163 if you are ready for this staff. – Moishe Kohan Oct 02 '23 at 15:27
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    On the positive side, every manifold is known to be homotopy-equivalent to a simplicial complex (if a manifold if compact, so is the complex). Thus, all homotopy-theoretic information can be encoded in a graph. See the references I gave in my answer here. – Moishe Kohan Oct 02 '23 at 15:30

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