An arithmetic analysis of closed surfaces

Abstract

From a computability-theoretic standpoint, we consider the following problem: Given a closed surface, as a topological space, how hard is it to recover an atlas? We prove that every computable Polish space homeomorphic to a closed surface admits an arithmetic atlas, and indeed an arithmetic triangulation. This is as simple as one could reasonably hope for; essentially, the locally Euclidean structure of a surface can be recovered from the topological structure in a first-order way, i.e., without reference to curves or homeomorphisms or other higher-order objects. It follows that given two computable presentations of the same closed surface, there is an arithmetic homeomorphism between them. Moreover, the homeomorphism problem for closed surfaces, presented as topological spaces, is arithmetic. From the algorithmic and definability-theoretic standpoint, this improves Kline’s conjecture proved by Bing in the 1940s. We also consider R 2 \mathbb {R}^2 and the closed unit ball.