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?



Are homeomorphic representations isomorphic?
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 ${\text{Z}}\_{2^r}$](https://doi.org/10.1090/S0273-0979-1990-15837-9), 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>.


