Good covers are algorithmically unrecognizable

TL;DR

This paper proves that determining whether a given simplicial complex is the nerve of a good cover in R^d is algorithmically undecidable for dimensions d ≥ 5, revealing fundamental limits in topological recognition problems.

Contribution

It establishes the undecidability of recognizing topologically d-representable complexes for fixed d ≥ 5, extending to acyclic covers and open d-balls, and relates PL embeddability to topological representability.

Findings

Recognition of topologically d-representable complexes is undecidable for d ≥ 5.

PL embeddability implies topological d-representability.

Subdivision conditions relate topological and PL embeddability.

Abstract

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable. More precisely, the intersection pattern of a good cover can be stored in a simplicial complex called nerve which records which subfamilies of the good cover intersect. A simplicial complex is topologically d-representable if it is isomorphic to the nerve of a good cover in R^d. We prove that it is algorithmically undecidable whether a given simplicial complex is topologically d-representable for any fixed d \geq 5. The result remains also valid if we replace good covers with acyclic covers or with covers by open d-balls. As an auxiliary result we prove that if a simplicial complex is PL embeddable into R^d, then it is topologically d-representable. We also supply this result with showing that if a "sufficiently fine" subdivision of a k-dimensional complex is d-representable and k \leq (2d-3)/3, then the complex is PL embeddable into R^d.