# 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}
}
• Published 31 May 2011
• Mathematics
• Journal of the ACM (JACM)
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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SODA 2012
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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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Discret. Comput. Geom.
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It is shown that the condition $\mathop{\mathrm{dim}}\nolimits X\leq 2k-1$ cannot be relaxed and the extension problem with (k−1)-connected Y becomes undecidable, and either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains Undecidable.
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