On the genus of representation spheres (Q917097)

For \(G=S^ 1\) and a \(G\)-representation \(V\), in order to prove the existence of critical points of \(G\)-invariant functionals from the representation sphere \(SV\) into the real numbers, \textit{V. Benci} [Commun. Pure Appl. Math. 34, 393-432 (1981; Zbl 0447.34040)] used the notion of geometrical index of the \(G\)-space in question. For a topological group \(G\) and a \(G\)-space \(X\), the author of the paper considers the following two versions (closely related to the index) of the \(G\)-genus of \(X\). The \(G\)-genus \(\gamma_ G(X)\) (resp. \({\tilde \gamma}{}_ G(X))\) is the minimal number \(n\) such that for some isotropy subgroups \(H_ 1,\dots,H_ n\) which occur in \(X\) (resp., some proper closed subgroups \(H_ 1,\dots,H_ n\) of \(G\)), there exists a continuous equivariant map from \(X\) into the join \(G/H_ 1*\dots*G/H_ n\) [cf. \textit{E. Fadell}, Pac. J. Math. 89, 33-42 (1980; Zbl 0464.55004)]. For a cyclic group \(G\) and a \(G\)-representation \(V\) without trivial summand, the main result of the paper (Theorem 1.2) describes lower bounds for \(\gamma_ G(SV)\); and when the order of \(G\) is a prime power, it also gives lower bounds for \({\tilde \gamma}{}_ G(SV)\). As a corollary, the author obtains lower bounds for the \(G\)-genus of lens spaces with free \(G\)-actions when \(G\) is the cyclic group of order 2 [cf. \textit{A. Pfister} and \textit{S. Stolz}, Comment. Math. Helv. 62, 286-291 (1987; Zbl 0634.10020)]. The proof of the main result is based on calculations in equivariant \(K\)-theory. The author provides up to date information about related results and he concludes the paper with some open problems and remarks about the equivariant Lusternik-Schnirelmann category \(cat_ G(X)\) related to the genus \(\gamma_ G(X)\) so that \(cat_ G(X)\geq \gamma_ G(X)\).