Computable structures on topological manifolds

TL;DR

This paper introduces a formal framework for defining and analyzing computable structures on topological manifolds, establishing foundational concepts and embedding theorems within computable topology.

Contribution

It develops a rigorous definition of computable manifolds using computable topology and Type-2 effectivity, including computable atlases and embeddings into Euclidean spaces.

Findings

Constructs a computable topological space from a given computable atlas.

Shows that compact computable manifolds satisfying a computable T2 axiom can be embedded into Euclidean space.

Provides a framework for computable submanifolds within the setting of computable topology.

Abstract

We propose a definition of computable manifold by introducing computability as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL manifolds respectively. Using the framework of computable topology and Type-2 theory of effectivity, we develop computable versions of all the basic concepts needed to define manifolds, like computable atlases and (computably) compatible computable atlases. We prove that given a computable atlas $\Phi$ defined on a set $M$, we can construct a computable topological space $(M, \tau_\Phi, \beta_\Phi, \nu_\Phi)$, where $\tau_\Phi$ is the topology on $M$ induced by $\Phi$ and that the equivalence class of this computable space characterizes the computable structure determined by $\Phi$. The concept of computable submanifold is also investigated. We show that any compact computable manifold which satisfies a computable version of the $T_2$-separation axiom, can be embedded as a computable submanifold of some euclidean space $\mathbb{R}^{q}$, with a computable embedding, where $\mathbb{R}^{q}$ is equipped with its usual topology and some canonical computable encoding of all open rational balls.