Extendability of Continuous Maps Is Undecidable

@article{adek2014ExtendabilityOC,
  title={Extendability of Continuous Maps Is Undecidable},
  author={Martin {\vC}adek and Marek Krc{\'a}l and Jir{\'i} Matousek and Luk{\'a}s Vokr{\'i}nek and Uli Wagner},
  journal={Discrete \& Computational Geometry},
  year={2014},
  volume={51},
  pages={24-66}
}
We consider two basic problems of algebraic topology: the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.The extension problem is the following: Given topological spaces X and Y, a subspace A⊆X, and a (continuous) map f:A→Y, decide whether f can be extended to a continuous map $\bar{f}\colon X\to Y$. All spaces are given as finite simplicial complexes, and the map f is simplicial.Recent positive algorithmic… 

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