Simplicial complex recognition problem

The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether it is homeomorphic to another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more.[1][2]: 9–11 

Background

An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC. Thus, an ASC provides a finite representation of a geometric object. Given an ASC, one can ask several questions regarding the topology of the GSC it represents.

Homeomorphism problem

The homeomorphism problem is: given two finite simplicial complexes representing smooth manifolds, decide if they are homeomorphic.

  • If the complexes are of dimension at most 3, then the problem is decidable. This follows from the proof of the geometrization conjecture.
  • For every d ≥ 4, the homeomorphism problem for d-dimensional simplicial complexes is undecidable.[3]

The same is true if "homeomorphic" is replaced with "piecewise-linear homeomorphic".

Recognition problem

The recognition problem is a sub-problem of the homeomorphism problem, in which one simplicial complex is given as a fixed parameter. Given another simplicial complex as an input, the problem is to decide whether it is homeomorphic to the given fixed complex.

  • The recognition problem is decidable for the 3-dimensional sphere .[4] That is, there is an algorithm that can decide whether any given simplicial complex is homeomorphic to the boundary of a 4-dimensional ball.
  • The recognition problem is undecidable for the d-dimensional sphere for any d ≥ 5. The proof is by reduction to the word problem for groups. From this, it can be proved that the recognition problem is undecidable for any fixed compact d-dimensional manifold with d ≥ 5.
  • As of 2014, it is open whether the recognition problem is decidable for the 4-dimensional sphere .[2]: 11 

Manifold problem

The manifold problem is: given a finite simplicial complex, is it homeomorphic to a manifold? The problem is undecidable; the proof is by reduction from the word problem for groups.[2]: 11 

References
  1. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN 9780387979700.
  2. Poonen, Bjorn (2014-10-25). "Undecidable problems: a sampler". arXiv:1204.0299 [math.LO].
  3. "A. Markov, "The insolubility of the problem of homeomorphy", Dokl. Akad. Nauk SSSR, 121:2 (1958), 218–220". www.mathnet.ru. Retrieved 2022-11-27.
  4. Matveev, Sergei (2003), Matveev, Sergei (ed.), "Algorithmic Recognition of S3", Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics, Berlin, Heidelberg: Springer, vol. 9, pp. 193–214, doi:10.1007/978-3-662-05102-3_5, ISBN 978-3-662-05102-3, retrieved 2022-11-27
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