Decidability of the Extension Problem for Maps into Odd-Dimensional Spheres

Abstract

In a recent paper (Cadek et al., Discrete Comput Geom 51:24---66, 2014), it was shown that the problem of the existence of a continuous map $$X \rightarrow Y$$XźY extending a given map $$A \rightarrow Y$$AźY, defined on a subspace $$A \subseteq X$$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 $$\pi _n Y$$źnY are finite for $$2d