Decidability of the extension problem for maps into odd-dimensional spheres

TL;DR

This paper proves that the problem of extending continuous maps into odd-dimensional spheres is decidable, contrasting with the undecidability result for even-dimensional spheres, and extends to certain $d$-connected target spaces.

Contribution

It establishes the decidability of the extension problem for maps into odd-dimensional spheres and related spaces, filling a gap in the understanding of this problem.

Findings

Decidability of the extension problem for odd-dimensional spheres.

Contrasts with previous undecidability results for even-dimensional spheres.

Applicable to $d$-connected spaces with finite homotopy groups.

Abstract

In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper, we prove that the same problem for $Y$ an odd-dimensional sphere is decidable. More generally, the same holds for any $d$-connected target space $Y$ whose homotopy groups $\pi_k Y$ are finite for $k>2d$.