On the genus of some subsets of $G$-spheres

The problem of estimating the genus of G-spaces (G-category) attracts considerable attention (see, for instance, [Bar1, Bar2, Fa, FaHu, Kr, LS, Šv] and others). At least two approaches to this problem exist: geometric, based on Borsuk–Ulam type theorems, and homological, based on (co)homological arguments in the study of orbit spaces. Historically, the first result concerning this problem is the famous Lusternik– Schnirelman Theorem stating that the category of the n-dimensional real projective space equals n + 1 (see [LS]). In terms of genus the Lusternik–Schnirelman Theorem can be formulated as follows: the genus of the n-dimensional sphere with respect to the antipodal action is equal to n + 1. This result was generalized by A. Fet [Fe] to the case of an arbitrary free involution on the sphere. The case of a free action of an arbitrary finite cyclic group was considered by M. Krasnosel’skĭı [Kr] in the framework of the geometric approach. A. Švartz [Šv] was the first to consider the case of a non-free action of a cyclic group on the sphere and obtained the following result: let the finite cyclic group Zp act on the n-dimensional unit sphere S, let A = {x ∈ S | ∃g ∈ Zp : g 6= 1 & gx = x}, and suppose dim A = k. Then gen(S \A) ≥ n− k, where gen(·) denotes genus.