Let $G$ be a finite group. Let $V_1, V_2$ be two finitedimensional 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?

2$\begingroup$ Maybe you assume the group is finite? Otherwise take $\mathbf{Z}$ acting on $\mathbf{R}$ by powers of $x\mapsto 2x$, resp. $x\mapsto 3x$. $\endgroup$– YCorCommented Apr 5, 2022 at 21:48

2$\begingroup$ This seems false for cyclic groups $\mathbb{Z}/n\mathbb{Z}$ acting on $\mathbb{C}$. There are conjugate characters. Their action is "the same up to conjugation," and so they are equivariantly homeomorphic, no? $\endgroup$– Geva YashfeCommented Apr 5, 2022 at 22:01

1$\begingroup$ @GevaYashfe Are they isomorphic as real representations? $\endgroup$– UVIRCommented Apr 5, 2022 at 22:05

1$\begingroup$ Sorry and thanks, I misread the question. $\endgroup$– Geva YashfeCommented Apr 5, 2022 at 22:07

1$\begingroup$ @LSpice $x \mapsto \operatorname{sign}(x) x^{ \log 3/\log 2}$. $\endgroup$– Will SawinCommented Apr 5, 2022 at 22:10
1 Answer
This is a famous problem, originating in work of de Rham, and the answer turns out to be No. The lowestdimensional examples of nonlinear 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.

1$\begingroup$ Are you by chance able to provide an explicit group (e.g., cyclic of order 8) and and explicit pair of nonequivalent representations that are topologically conjugate? $\endgroup$– YCorCommented Apr 6, 2022 at 18:18

1$\begingroup$ In principle I think one can work out an explicit example, e.g. for the cyclic group of order 8, from the information in Cappell and Shaneson's announcement projecteuclid.org/journals/… $\endgroup$ Commented Apr 6, 2022 at 18:49

1$\begingroup$ I hope someone will work out some such example and post it... $\endgroup$ Commented Apr 6, 2022 at 18:50

6$\begingroup$ Oh, it's explicit in the original CappellShaneson paper. They provide many explicit examples, one of which (p318 pick $q=2$, $j=k=1$), for the cyclic group of order 8, is as follows: define $r_\theta$ as the rotation matrix of angle $2\pi\theta/8$ is given by the two 9dimensional real representations given by powers of blockdiagonal matrices $(r_1,r_1,r_1,r_1,1)$ and $(r_5,r_5,r_5,r_5,1)$. $\endgroup$– YCorCommented Apr 6, 2022 at 19:11
