The simplicial complex recognition problem is a

computational problem In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computati ...

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. Given a

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

, the problem is to decide whether it is

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more.

Background

abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...

(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 In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their ''n''-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in th ...

(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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topology ...

, decide if they are

. * If the complexes are of dimension at most 3, then the problem is decidable. This follows from the proof of the

geometrization conjecture In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theor ...

. * For every d ≥ 4, the homeomorphism problem for ''d''-dimensional simplicial complexes is undecidable. 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 A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its ...

. 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

? The problem is undecidable; the proof is by reduction from the

References

{{reflist Undecidable problems Simplicial sets