I'm given two simplicial sets $X,Y : \Delta^{\operatorname{op}} \to Set$. Of, course, if I study their geometric realisations $\lvert X \rvert$ and $\lvert Y \rvert$, I might find a homeomorphism, or maybe I can show via homology that they aren't homeomorphic. But how do I find that out just given the combinatorial data? Is there a combinatorial condition? Or is there an algorithm that can decide it for simplicial sets with finitely many nondegenerate simplices?

Edit: Is there a set of moves (similar to Pachner moves for triangulations) that can be applied to go from one simplicial set to another?

  • $\begingroup$ Weibel, in An Introduction to Homological Algebra, talks about exactly this in the chapter "Simplicial methods in homological algebra". $\endgroup$ – Andy May 14 '15 at 17:32
  • $\begingroup$ @Andy Really? I can only find homotopy equivalence, but not homeomorphism. $\endgroup$ – Turion May 14 '15 at 18:03
  • $\begingroup$ Oh, I must have misread the question, sorry. I hope it was worth the read though :) $\endgroup$ – Andy May 14 '15 at 19:06

There is no such algorithm. By a famous theorem of Markov, there is in fact no algorithm that takes two triangulations of compact $4$-manifolds and decides whether the manifolds are homeomorphic (you can replace $4$ with any $n\geq 4$ here as well). The idea is that given a finite presentation of a group, you can computably describe a $4$-manifold with that group as its fundamental group, such that the manifolds are homeomorphic iff the groups are isomorphic. Since there is no algorithm that can decide if two finite presentations describe isomorphic groups, there can be no algorithm that decides from a triangulation whether manifolds are homeomorphic. (Actually, I'm not sure if this is exactly how the argument goes, but it's something similar.)

  • $\begingroup$ OK, asking for an algorithm is too strong. Is there some statement like "the simplicial sets can be related by the following moves", and for moves insert something generalising Pachner moves for triangulated manifolds? $\endgroup$ – Turion Jan 14 '16 at 12:59

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