2013/02/10 by Martin Cadek, Marek Krcal, Cadek, Martin +7
Computer Science · Mathematics · #55P99 #55Q05 #55S35 #55S36 #68Q17 #68U05 #68W99 #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #cs.CG #math.AT #msc:55P99 #msc:55Q05 #msc:55S35 #msc:55S36 #msc:68Q17 #msc:68U05 #msc:68W99
paper · pdf · doi:10.48550/arxiv.1302.2370
38 pages
arxiv created 2013/02/10 · arxiv updated 2013/02/12
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⊆ X, and a (continuous) map f:A->Y, decide whether f can be extended to a continuous map 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 (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 dim X<=2k-1 cannot be relaxed: for dim 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.