Discrete & Computational Geometry

, Volume 51, Issue 1, pp 24–66 | Cite as

Extendability of Continuous Maps Is Undecidable

  • Martin Čadek
  • Marek Krčál
  • Jiří Matoušek
  • Lukáš Vokřínek
  • Uli Wagner
Article

Abstract

We consider two basic problems of algebraic topology: the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.

The extension problem is the following: Given topological spaces X and Y, a subspace AX, and a (continuous) map f:AY, decide whether f can be extended to a continuous map \(\bar{f}\colon X\to Y\). All spaces are given as finite simplicial complexes, and the map f is simplicial.

Recent positive algorithmic results, proved in a series of companion papers, show that for (k−1)-connected Y, k≥2, the extension problem is algorithmically solvable if the dimension of X is at most 2k−1, and even in polynomial time when k is fixed.

Here we show that the condition \(\mathop{\mathrm{dim}}\nolimits X\leq 2k-1\) cannot be relaxed: for \(\mathop{\mathrm{dim}}\nolimits X=2k\), the extension problem with (k−1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains undecidable.

Our second result, a strengthening of a result of Anick, says that the computation of πk(Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Martin Čadek
    • 1
  • Marek Krčál
    • 2
  • Jiří Matoušek
    • 2
    • 3
  • Lukáš Vokřínek
    • 1
  • Uli Wagner
    • 4
  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  3. 3.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  4. 4.Institut de Mathématiques de Géométrie et Applications, École Polytechnique Fédérale de LausanneEPFL SB MATHGEOMLausanneSwitzerland

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