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 ⊆ X, and a (continuous) map f : A → Y , decide whether f can be extended to a continuous mapf : 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 π k (Y ) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

Computing All Maps into a Sphere

“…, k. Equivalently, Y is path-connected and the first k homotopy groups πi(Y ), i ≤ k, are trivial. 7 Steenrod [53] calls this restriction "most severe," and conjectures that it "should ultimately be unnecessary." encountered in applications, is the sphere S d .…”

“…The dimension and connectivity assumptions in Theorem 1.1 turned out to be essential and almost sharp, in the following sense: In [7], it is shown that, for every d ≥ 2, the extension problem is undecidable for dim X = 2d and Dependence on d. The running-time of the algorithm in Theorem 1.1 can be made polynomial for every fixed d, as was mentioned above, but it depends on d at least exponentially. We consider it unlikely that the problem can be solved by an algorithm whose running time also depends polynomially on d. One heuristic reason supporting this belief is that Theorem 1.1 includes the computation of the stable homotopy groups π d+k (S d ), k ≤ d − 2.…”

“…These are considered mathematically very difficult objects, and a polynomial-time algorithm for computing them would be quite surprising. Another reason is that the related problem of computing the higher homotopy groups π k (Y ) of a 1-connected simplicial complex Y was shown to be #P-hard if k, encoded in unary, is a part of input [1,7], and it is W[1]-hard w.r.t. the parameter k [32], even for Y of dimension 4.…”

Extending continuous maps

“…We devise another way of converting Anick's Y into a simplicial complex, which produces only polynomially many simplices. 4 The idea is simple, but there are several technical issues to be worked out; we refer to [4] for the derivation of Theorem 1.2 from Anick's result.…”

“…This survey summarizes the results of the three papers [12,5,4]. Together these comprise well over 100 pages, and they deal with numerous moderately advanced topological concepts, as well as many algorithmic notions and results.…”