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Discrete & Computational Geometry

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

Algorithmic Solvability of the Lifting-Extension Problem

  • Martin Čadek
  • Marek Krčál
  • Lukáš Vokřínek
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 55Q05 Secondary 55S91 

Notes

Acknowledgements

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].

References

  1. 1.
    Bredon, G.E.: Equivariant Cohomology Theories. Lecture Notes in Mathematics, vol. 34. Springer, Berlin (1967)Google Scholar
  2. 2.
    Brown, E.H.: Finite computability of Postnikov complexes. Ann. Math. 65, 1–20 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Čadek, M., Krčál, M., Vokřínek, L.:  Algorithmic solvability of the lifting-extension problem (extended version). http://arxiv.org/abs/1307.6444 (2013)
  4. 4.
    Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., Wagner, U.: Computing all maps into a sphere. J. ACM 61(17), 1–44 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM J. Comput. 43, 1728–1780 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Extendability of continuous maps is undecidable. Discrete Comput. Geom. 51, 24–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crabb, M., James, I.: Fibrewise Homotopy Theory. Springer Monographs in Mathematics. Springer, London (1998)Google Scholar
  8. 8.
    Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 73–126. Elsevier, Amsterdam (1995)CrossRefGoogle Scholar
  9. 9.
    Eilenberg, S., MacLane, S.: On the groups \(H(\Pi,n)\), II. Methods of computation. Ann. Math. 60, 49–139 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Filakovský, M.: Effective chain complexes for twisted products. Arch. Math. (Brno) 48, 313–322 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Filakovský, M.: Algorithmic construction of the Postnikov tower for diagrams of simplicial sets. Thesis. https://is.muni.cz/th/211334/prif_d/THESIS_Marek_Filakovsky.pdf
  12. 12.
    Filakovský, M., Vokřínek, L.: Are two given maps homotopic? An algorithmic viewpoint. http://arxiv.org/abs/1312.2337 (2013)
  13. 13.
    Friedman, G.: An elementary illustrated introduction to simplicial sets. Rocky Mt. J. Math. 42, 353–423 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence, RI (2003)Google Scholar
  15. 15.
    Matoušek, J.: Using the Borsuk–Ulam Theorem. Springer, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Matoušek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in \({\mathbb{R}}^{d}\). J. Eur. Math. Soc. 13, 259–295 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    May, J.P.: Simplicial Objects in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1992)Google Scholar
  18. 18.
    McClendon, J.F.: Obstruction theory in fiber spaces. Math. Z. 120, 1–17 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robinson, C.A.: Moore-Postnikov systems for non-simple fibrations. Ill. J. Math. 16, 234–242 (1972)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rubio, J., Sergeraert, F.: Constructive homological algebra and applications. http://arxiv.org/abs/1208.3816 (2012) (Written in 2006 for a MAP Summer School at the University of Genova)
  21. 21.
    Shih, W.: Homologie des espaces fibres. Publ. Math. l’IHÉS 13, 93–176 (1962)zbMATHGoogle Scholar
  22. 22.
    Stasheff, J.: \(H\)-Spaces From a Homotopy Point of View. Lecture Notes in Mathematics, vol. 161. Springer, Berlin (1970)Google Scholar
  23. 23.
    Vokřínek, L.: Computing the abelian heap of unpointed stable homotopy classes of maps. Arch. Math. (Brno) 49, 359–368 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vokřínek, L.: Constructing homotopy equivalences of chain complexes of free \({\mathbb{Z}}G\)-modules. Contemp. Math. 617, 279–296 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vokřínek, L.: Heaps and unpointed stable homotopy theory. Arch. Math. (Brno) 50, 323–332 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vokřínek, L.: Decidability of the extension problem for maps into odd-dimensional spheres. Discrete Comput. Geom. 57(1), 1–11 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weber, C.: Plongements de polyedres dans le domaine metastable. Comment. Math. Helv. 42, 1–27 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Martin Čadek
    • 1
  • Marek Krčál
    • 2
    • 3
  • Lukáš Vokřínek
    • 1
  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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