On the stable topological similarity of representations and their restrictions to cyclic subgroups (Q1069514)

Linear representations \(\rho_ 1,\rho_ 2\) of a finite group G on real vector spaces \(V_ 1,V_ 2\), respectively, are called topologically similar if there is a topological homeomorphism \(\phi\) : \(V_ 1\to V_ 2\) such that \(\phi \circ \rho_ 1(g)=\rho_ 2(g)\circ \phi\) for all \(g\in G\). \textit{S. E. Cappell} and \textit{J. L. Shaneson} [Am. J. Math. 104, 773-778 (1982; Zbl 0497.57007)] have asked the following question: If \(\rho_ 1,\rho_ 2\) are linear representations of G and \(\rho_ 1| C\) is topologically similar to \(\rho_ 2| C\) for all cyclic subgroups C of G, is there a linear representation \(\theta\) of G such that \(\rho_ 1+\theta\) is topologically similar to \(\rho_ 2+\theta ?\) In the present note it is shown that this question has a negative answer.