Decidability of the Extension Problem for Maps into Odd-Dimensional Spheres
In a recent paper (Čadek et al., Discrete Comput Geom 51:24–66, 2014), it was shown that the problem of the existence of a continuous map \(X \rightarrow Y\) extending a given map \(A \rightarrow Y\), 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 \(\pi _n Y\) are finite for \(2d<n<\dim X\). We also prove an equivariant version, where all spaces are equipped with free actions of a given finite group G and all maps are supposed to respect these actions. This yields the computability of the \(\mathbb Z/2\)-index of a given space up to an uncertainty of 1.
KeywordsHomotopy class Computation Higher difference
Mathematics Subject ClassificationPrimary 55Q05 Secondary 55S35
I am very grateful to the reviewer for his/her useful comments and suggestions. In particular, the reviewer pointed out that the construction of the actions on Postnikov stages (carried out in an earlier version) is completely unnecessary and that the mere existence of actions is sufficient; this has led to many simplifications throughout the paper. The research was supported by the Center of Excellence, Eduard Čech Institute (Project P201/12/G028 of GA ČR).
- 4.Čadek, M., Krčál, M., Vokřínek L.: Algorithmic solvability of the lifting-extension problem. Preprint, arXiv:1307.6444 (2013)
- 5.Franek, P., Krčál, M.: Robust satisfiability of systems of equations. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 193–203 (2014)Google Scholar
- 6.Filakovský, M., Vokřínek, L.: Are two given maps homotopic? An algorithmic viewpoint. Preprint, arXiv:1312.2337 (2013)