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The Unsolvability of the Homeomorphism Problem

Stefan Friedl, Tobias Hirsch, Marc Kegel

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.

The Unsolvability of the Homeomorphism Problem

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.
Paper Structure (4 sections, 11 theorems)

This paper contains 4 sections, 11 theorems.

Table of Contents

  1. Introduction
  2. From group presentations to PL-simplicial manifolds
  3. Unsolvability of the decision problem
  4. Unrecognizable manifolds

Key Result

Theorem 1.1

For all integers $n\geq4$ there exists no algorithm that

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 8 more