Equivariant \(K\)-theoretic Euler classes and maps of representation spheres (Q5937411)

Let \(G\) be a compact Lie group and \(U\) a unitary representation of \(G\). Denote by \(SU\) the representation sphere of unit length in \(U\). By using equivariant \(K\)-theory, the author shows that if there exists a \(G\)-map \(f:SU \to SW\) where \(W\) is another representation of \(G\), then there is an element \(z(f)\) in the complex representation ring \(R(G)\) such that \(e(W)=z(f)\cdot e(U)\) where \(e(Z)\) denotes the Euler class in \(K_G(pt.)=R(G)\). Results concerning the topological degree of a \(G\)-map, existence of \(G\)-maps between representation spheres, and the non-vanishing of the \(K\)-theoretic Euler class are given.