A class of open surfaces with algorithmically solvable homeomorphism problem

TL;DR

This paper introduces topological automata, a broad class of open manifolds, and proves that the homeomorphism problem is decidable for 2-dimensional cases, advancing understanding in low-dimensional topology.

Contribution

It defines topological automata and establishes the decidability of the homeomorphism problem for 2-dimensional open manifolds within this class.

Findings

Homeomorphism problem is decidable for 2D automata.

The class includes many interesting open 2D and 3D manifolds.

Provides a framework for algorithmic analysis of open manifolds.

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.