Extendability of continuous maps is undecidable
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 A\subseteq X, and a (continuous) map f:A>Y, decide whether f can be extended to a continuous map \bar{f}:X>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 (k1)connected Y, k>=2, the extension problem is algorithmically solvable if the dimension of X is at most 2k1, and even in polynomial time when k is fixed. Here we show that the condition \dim X<=2k1 cannot be relaxed: for \dim X=2k, the extension problem with (k1)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 \pi_k(Y) of a 1connected simplicial complex Y is #Phard when k is considered as a part of the input.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 arXiv:
 arXiv:1302.2370
 Bibcode:
 2013arXiv1302.2370C
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Algebraic Topology;
 68U05;
 68W99;
 68Q17;
 55S35;
 55S36;
 55P99;
 55Q05
 EPrint:
 38 pages