Rational homotopy type and computability

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.

26 pages. This is a major revision: The former Lemma 7.2 had been proven incorrectly and is now a conjecture, as is one direction of what was previously the main theorem. I have added proofs of a number of special cases as well as an explanation of why the general statement seems very difficult

References in corpus (2)

• Integral and rational mapping classes
• Fibrewise rational H-spaces

Cited by in corpus (1)

• Homotopy Spectra and Diophantine Equations