Non-Linear Similarity Revisited
Ian Hambleton, Erik Kjær Pedersen · Princeton University Press eBooks · 1996
Let G be a finite group and V, V ′ finite dimensional real orthogonal representations of G. Then V is said to be topologically similar to V ′ (V ∼t V ′) if there exists a homeomor-phism h: V → V ′ which is G-equivariant. If V, V ′ are topologically similar, but not linearly isomorphic, then such a homeomorphism is called a non-linear similarity. The topological classification of G-representations was first studied by de Rham [18]. He proved that if a topological similarity h: V → V ′ of orthogonal representations preserves the unit spheres and restricts to a diffeomorphism between S(V) and S(V ′), then V and V ′ are linearly isomorphic. In 1973, Kuiper and Robbin [11] obtained positive results on the general problem and conjectured that topological equivalence implies linear equivalence for all finite groups G. However, in 1981 Cappell and Shaneson [1] constructed the first examples of non-linear similarities. The simplest occurs for G = Z/8, but they also constructed a large class of examples for cyclic groups of the form G = Z/4q. Further results can be found in [2], [3], [4], and [13]. On the other hand, Hsiang and Pardon [10] and Madsen and Rothenberg [12] indepen-dently proved the conjecture for all odd–order groups. In addition, the main theorem of [10]