Decidability of the Extension Problem for Maps into Odd-Dimensional Spheres
| Authors | |
|---|---|
| Year of publication | 2017 |
| Type | Article in Periodical |
| Magazine / Source | Discrete & Computational Geometry |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.1007/s00454-016-9835-x |
| Field | General mathematics |
| Keywords | Homotopy class; Computation; Higher difference |
| Description | 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 -> Y extending a given map A -> Y, defined on a subspace A subset of 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 Z/2-index of a given space up to an uncertainty of 1. |
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