  We will describe some results regarding the algorithmic nature of
homeomorphism problems for manifolds; in particular, the following theorem.
  Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at
least 5 can be recognized among simply connected manifolds. That is, there is
an algorithm to decide whether or not another simply connected manifold is Top,
PL or Diff isomorphic to M. Moreover, an analogous statement is true for
embeddings in codimension at least three: one can algorithmically recognize any
given embedding of one simply connected manifold in another up to isomorphism
of pairs, or up to isotopy, if the codimension of the embedding is not two.
