Computing All Maps into a Sphere

@article{adek2014ComputingAM,
  title={Computing All Maps into a Sphere},
  author={Martin {\vC}adek and Marek Krc{\'a}l and Jir{\'i} Matousek and Francis Sergeraert and Luk{\'a}s Vokr{\'i}nek and Uli Wagner},
  journal={Journal of the ACM (JACM)},
  year={2014},
  volume={61},
  pages={1 - 44}
}
Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of such maps. We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d−2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical… 
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Computing all maps into a sphere
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An algorithm for computing all homotopy classes of continuous maps X → Y, where X, Y are topological spaces given as finite simplicial complexes, which guarantees that [X, Y] has a natural structure of a finitely generated Abelian group, and the algorithm finds generators and relations for it.
Extendability of Continuous Maps Is Undecidable
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