Sylvain Cappell was here in Madison today talking about classification of group actions on manifolds, and he mentioned a crazy fact, due to himself and Shaneson, that I didn’t know. You can have two matrices, i.e. linear transformations of R^d, which are not conjugate in GL_d(R), but they are conjugate by some homeomorphism of R^d! That’s cool. And to me kind of amazing! People who know the paper are welcome to explain in comments why I should not have been amazed by this fact, if that’s indeed the case.
Update: As a couple of people point out in comments, this fact is really not so amazing as I stated it, because e.g. multiplication by 5 and multiplication by 125 on R^1 are conjugate by x -> x^3! Looking at Cappell and Shaneson’s “Non-linear similarity,” I see that the real point is to prove this for matrices with eigenvalues on the unit circle. This makes the eigenvalues “feel” much more like topological invariants, in the sense that there are certain sequences of powers of the matrix that move closer and closer to the identity. It’s a theorem of Poincare (I learn from C&S) that 2×2 matrices with norm-1 eigenvalues are topologically conjugate if and only if they’re linearly conjugate. And de Rham showed (at least for orthogonal matrices) that topological conjugacy can’t change any eigenvalue that’s not a root of unity. Then Nicolaas Kuiper and my UW colleague Joel Robbin extended this to the general linear group, and conjectured that topological conjugacy implied conjugacy in general. Lots of cases of the Kuiper-Robbin conjecture were proved, e.g. by Sullivan and Schwartz; for instance it is true for matrices of odd prime power order. So what Cappell and Shaneson did was construct the first counterexamples to the Kuiper-Robbin conjecture. And more: they go a long way towards classification of linear transformations with norm-1 eigenvalues up to topological conjugacy, showing e.g. that the two notions agree in dimensions up to 5.
By the way, as Tom Graber pointed out in comment, you can’t make two non-conjugate linear maps conjugate via a diffeomorphism, because you can read the eigenvalues off the action on the tangent space at 0. But Capell and Shaneson show that you can get the job done with a homeomorphism that’s smooth everywhere except o! So the obvious obstruction is in some sense the only one.