Rational homotopy type and computability

TL;DR

This paper investigates the algorithmic decidability of extending maps in simplicial pairs, showing decidability when the target space has a rational homotopy type of an H-space, and linking other cases to Hilbert's tenth problem.

Contribution

It establishes a clear criterion for decidability based on the rational homotopy type of the target space, connecting algebraic topology with computability theory.

Findings

Decidability for extensions when Y has the rational homotopy type of an H-space.

Undecidability results for other target spaces Y.

Implications for bundle structure questions over finite complexes.

Abstract

Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, does $f$ have an extension to $X$? We show that for a fixed $Y$, this question is algorithmically decidable for all $X$, $A$, and $f$ if $Y$ has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other $Y$, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable.