simplicial complex recognition problem
{{Short description|Computational problem in algebraic topology}}
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.{{citation |last=Stillwell |first=John |title=Classical Topology and Combinatorial Group Theory |url=https://books.google.com/books?id=265lbM42REMC&pg=PA247 |volume=72 |page=247 |year=1993 |series=Graduate Texts in Mathematics |publisher=Springer |isbn=9780387979700 |authorlink=John Stillwell}}.{{cite arXiv |last=Poonen |first=Bjorn |date=2014-10-25 |title=Undecidable problems: a sampler |class=math.LO |eprint=1204.0299 }}{{Rp|pages=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.{{Cite web |title=A. Markov, "The insolubility of the problem of homeomorphy", Dokl. Akad. Nauk SSSR, 121:2 (1958), 218–220 |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=23250&option_lang=eng |access-date=2022-11-27 |website=www.mathnet.ru}}
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 .{{Citation |last=Matveev |first=Sergei |title=Algorithmic Recognition of S3 |date=2003 |url=https://doi.org/10.1007/978-3-662-05102-3_5 |work=Algorithmic Topology and Classification of 3-Manifolds |series=Algorithms and Computation in Mathematics |volume=9 |pages=193–214 |editor-last=Matveev |editor-first=Sergei |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-05102-3_5 |isbn=978-3-662-05102-3 |access-date=2022-11-27|url-access=subscription }} 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 from 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 .{{Rp|pages=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.{{Rp|pages=11}}