This is one of the few problems from the Scottish book that are still open. In slightly modernized form, and correcting the typo (in the book, and
should be switched in the last sentence) the problem is:
Let
be a topological manifold, and let
be a continuous function. Let
denote the subgroup of homeomorphisms
that satisfy
. Let
be another manifold that is not homeomorphic to
. Does there exist a continuous function
such that
is not isomorphic to
?