Discrete & Computational Geometry

, Volume 57, Issue 4, pp 915–965 | Cite as

Algorithmic Solvability of the Lifting-Extension Problem

  • Martin Čadek
  • Marek Krčál
  • Lukáš VokřínekEmail author


Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and \(\dim X\le 2d\), for some \(d\ge 1\), we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps \(|X|\rightarrow |Y|\); the existence of such a map can be decided even for \(\dim X\le 2d+1\). This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into \(\mathbb R^n\) under the condition \(k\le \frac{2}{3} n-1\). More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.


Homotopy classes Equivariant Fibrewise Lifting-extension problem Algorithmic computation Embeddability Moore–Postnikov tower 

Mathematics Subject Classification

Primary 55Q05 Secondary 55S91 



We are grateful to Jiří Matoušek and Uli Wagner for many useful discussions, comments and suggestions that improved this paper a great deal. Moreover, this paper could hardly exist without our long-term collaboration, partly summarized in [4, 5, 6].


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Martin Čadek
    • 1
  • Marek Krčál
    • 2
    • 3
  • Lukáš Vokřínek
    • 1
    Email author
  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic
  2. 2.Computer Science Institute of Charles UniversityPrague 1Czech Republic
  3. 3.Institute of Science and Technology AustriaKlosterneuburgAustria

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