research-article

Computing All Maps into a Sphere

Authors:

Jiří Matoušek,

Francis Sergeraert,

Lukáš Vokřínek,

Uli WagnerAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 3

Article No.: 17, Pages 1 - 44

https://doi.org/10.1145/2597629

Published: 02 June 2014 Publication History

Abstract

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 S^d. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X=S¹ it includes a well known undecidable problem: testing triviality of the fundamental group of Y.

In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A → Y and ask whether it extends to a map X → Y, or computing the ℤ₂-index—everything in the stable range. Outside the stable range, the extension problem is undecidable.

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Index Terms

Computing All Maps into a Sphere
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image Journal of the ACM

Journal of the ACM Volume 61, Issue 3

May 2014

262 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2628069

Issue’s Table of Contents

Copyright © 2014 ACM.

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Publication History

Published: 02 June 2014

Accepted: 01 February 2014

Revised: 01 January 2014

Received: 01 September 2011

Published in JACM Volume 61, Issue 3

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Czech Ministry of Education
European Research Council
project GAUK 49209
project 1M0545 by the Ministry of Education of the Czech Republic and by Center of Excellence -- Institute for Theoretical Computer Science
Swiss National Science Foundation
Prague (project P202/12/G061 of GAČR)

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https://doi.org/10.1007/s00454-023-00513-0
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Ellis GKilleen K(2021)Cohomology with local coefficients and knotted manifoldsJournal of Symbolic Computation10.1016/j.jsc.2021.04.004107(299-321)Online publication date: Nov-2021
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