Simplicial complex recognition problem explained

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]}

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

S ^{3} . 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

S ^{d} 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

S ^{4} .

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.

Notes and References

1. .
2. Poonen . Bjorn . 2014-10-25 . Undecidable problems: a sampler . math.LO . 1204.0299 .
3. Web site: A. Markov, "The insolubility of the problem of homeomorphy", Dokl. Akad. Nauk SSSR, 121:2 (1958), 218–220 . 2022-11-27 . www.mathnet.ru.