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A class of open surfaces with algorithmically solvable homeomorphism problem

Sylvain Maillot

Abstract

We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and 3-dimensional manifolds of interest to low-dimensional topologists. Our main result is that the homeomorphism problem in this class is decidable for n = 2.

A class of open surfaces with algorithmically solvable homeomorphism problem

Abstract

We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and 3-dimensional manifolds of interest to low-dimensional topologists. Our main result is that the homeomorphism problem in this class is decidable for n = 2.

Paper Structure

This paper contains 35 sections, 13 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. From automata to admissible trees
  4. Decorated graphs and admissible trees
  5. From planar 2-automata to decorated graphs
  6. From decorated graphs to admissible trees
  7. Step 1
  8. Step 2
  9. Step 3
  10. Classification of admissible trees
  11. Reduced admissible trees
  12. Move 1
  13. Move 2
  14. Move 3
  15. Decomposing a totally disconnected space
  16. ...and 20 more sections

Key Result

Theorem 1.5

There is an algorithm which takes as input two topological $2$-automata $\mathcal{X}_1,\mathcal{X}_2$ and decides whether $M(\mathcal{X}_1)$ and $M(\mathcal{X}_2)$ are homeomorphic.

Figures (7)

  • Figure 1: A topological $2$-automaton $\mathcal{X}$
  • Figure 2: The surface associated to $\mathcal{X}$
  • Figure 3: Making the graph $G$ into a tree
  • Figure 4: Making the tree admissible
  • Figure 5: Move 1
  • ...and 2 more figures

Theorems & Definitions (45)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Theorem 1.5
  • Theorem 1.6: Classification of surfaces, Kerékjártó, Richards richards:noncompact
  • Example 1.7
  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • ...and 35 more