Equivariant maps between representation spheres of cyclic \({p}\)-groups (Q1687135)

Let \(V, W\) be unitary representations of the cyclic group \(G\) of prime power order \(p^N\). It is assumed that the fixed point sets \(V^G=\{0\}=W^G\) are trivial, so that \(G\) acts without fixed points on the unit spheres \(S(V)\), \(S(W)\) of \(V\) and \(W\), respectively. It has been proved in [\textit{T. Bartsch}, Comment. Math. Helv. 65, No. 1, 85--95 (1990; Zbl 0704.57024)] that if there exists an equivariant map \(S(V)\to S(W)\) then \(p^{N-1}(\dim(V)-1)-(\dim(W)-1)\geq 0\). In the present paper this result is improved and refined by taking the isotypical decompositions of \(V\) and \(W\) into account. The proof is based on equivariant \(K\)-theory.