A linearity theorem for group actions on spheres with applications to homotopy representations
Let G be a finite group and \(T\subset {\mathbb{Q}}\) denote a subring which is contained in the local ring \({\mathbb{Z}}_{(p)}\) for any prime divisor p of the order of G. The author of the paper shows that if X is a finitistic G-space which is a T-homology sphere, then there exists a virtual representation V-W in the ring R(G) such that for any prime power order subgroup H of G, dim \(X^ H=\dim V^ H-\dim W^ H\). Then he applies this to give a complete description of the Grothendieck group of dimension functions of homotopy representations for compact Lie groups G. The description is provided in terms of dimension functions of linear representations as well as in combinatorial terms. In special cases, related results were obtained by \textit{R. M. Dotzel} and \textit{G. C. Hamrick} [Invent. Math. 62, 437-442 (1980; Zbl 0453.57030)] and \textit{T. tom Dieck} [Invent. Math. 67, 231-252 (1982; Zbl 0507.57026)].
dimension functions of homotopy representations for compact Lie groups
scientific article; zbMATH DE number 4107758