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 ?