TITLE: Are homeomorphic representations isomorphic?
QUESTION [29 upvotes]: Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are isomorphic representations?

REPLY [42 votes]: This is a famous problem, originating in work of de Rham, and the answer turns out to be No. The lowest-dimensional examples of non-linear similarity, as it is called, are in dimension 6, and examples only exist if the group has order divisible by (but not equal to) 4. This article contains a summary of the subject:
Sylvain Cappell, Julius Shaneson, Mark Steinberger, Shmuel Weinberger, and James West. The classification of nonlinear similarities over ${\text{Z}}_{2^r}$, Bulletin of the American Mathematical Society 22 (1990).
Mark Steinberger wrote another short summary of the subject, available here: http://math.albany.edu/topics/steinberger/msteinbergerrsch.pdf.