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

In a recent paper [3], it was shown that the problem of existence of a continuous map X → Y extending a given map A → Y defined on a subspace A ⊆ 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 π k Y are finite for k > 2d.