  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 (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 \pi_k(Y) of
a 1-connected simplicial complex Y is #P-hard when k is considered as a part of
the input.
