The Unsolvability of the Homeomorphism Problem

2026-03-24

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this short expository note, we give a detailed proof of Markov's theorem on the unsolvability of the homeomorphism problem and of the existence of unrecognizable manifolds in all dimensions larger than 3.

Problem

• homeomorphism problem
• unsolvability
• unrecognizable manifolds
• Markov's theorem

Innovation

• unsolvability
• homeomorphism problem
• unrecognizable manifolds
• Markov's theorem
• computational topology