The world problem: on the computability of the topology of 4-manifolds

2005/06/03 by James R. van Meter, Ernest Gatlin, van Meter, James R. +1
Computer Science · Physics and Astronomy · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Topological and Geometric Data Analysis #gr-qc

paper · pdf · doi:10.48550/arxiv.gr-qc/0506019

Submitted to Class. Quant. Grav

arxiv created 2005/06/03 · openalex publication_date 2005/06/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and only if the corresponding Turing machine halts on the associated input. Physical implications are briefly discussed.

Related

• Algorithmic aspects of homeomorphism problems
• Algorithms in 4-manifold topology
• On the homeomorphism problem for 4-manifolds
• Simpler algorithmically unrecognizable 4-manifolds
• A survey of the foundations of four-manifold theory in the topological category