# Are homeomorphic representations isomorphic?

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?

• Maybe you assume the group is finite? Otherwise take $\mathbf{Z}$ acting on $\mathbf{R}$ by powers of $x\mapsto 2x$, resp. $x\mapsto 3x$.
– YCor
Apr 5, 2022 at 21:48
• 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? Apr 5, 2022 at 22:01
• @GevaYashfe Are they isomorphic as real representations?
– UVIR
Apr 5, 2022 at 22:05
• Sorry and thanks, I misread the question. Apr 5, 2022 at 22:07
• @LSpice $x \mapsto \operatorname{sign}(x) |x|^{ \log 3/\log 2}$. Apr 5, 2022 at 22:10

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.

• Are you by chance able to provide an explicit group (e.g., cyclic of order 8) and and explicit pair of non-equivalent representations that are topologically conjugate?
– YCor
Apr 6, 2022 at 18:18
• 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/… Apr 6, 2022 at 18:49
• I hope someone will work out some such example and post it... Apr 6, 2022 at 18:50
• Oh, it's explicit in the original Cappell-Shaneson 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 9-dimensional real representations given by powers of block-diagonal matrices $(r_1,r_1,r_1,r_1,-1)$ and $(r_5,r_5,r_5,r_5,-1)$.
– YCor
Apr 6, 2022 at 19:11
• Thanks! This is very surprising.
– UVIR
Apr 7, 2022 at 21:30