The surjection property and computable type

TL;DR

This paper investigates the surjection and epsilon-surjection properties of spaces related to computable type, developing techniques to analyze these properties and applying them to finite simplicial complexes, including an answer to an open question.

Contribution

It introduces methods using homotopy and homology to study these properties and shows computable type is not preserved under products, also proving its decidability for finite simplicial complexes.

Findings

Computable type is not preserved under taking products.

Decidability of computable type for finite simplicial complexes.

Development of techniques using homotopy and homology theories.

Abstract

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the epsilon-surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes.