Simplicial complex recognition problem
From Wikipedia, the free encyclopedia
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
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".
