Extendability of Continuous Maps Is Undecidable

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 AaS dagger X, and a (continuous) map f:A -> Y, decide whether f can be extended to a continuous map . 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, ka parts per thousand yen2, 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 cannot be relaxed: for , 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 pi (k) (Y) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.