On the genus of some subsets of \(G\)-spheres (Q1899885)

In the work of \textit{A. S. Shvarts (Schwarz)} [Am. Math. Soc., Transl., II. Ser., 55, 49-140 (1966); translation from Tr. Mosk. Mat. O-va. 10, 217-272 (1961; Zbl 0178.26202)] in terms of the genus the following result was obtained: Let the finite cyclic group \(\mathbb{Z}_p\) act on the \(n\)-dimensional unit sphere \(S^n\), let \(A= \{x\in S^n|\exists g\in \mathbb{Z}_p: g\neq 1, gx= x\}\), and \(\dim A= k\); then \(\text{gen}(S^n\backslash A)\geq n- k\). In the article under review the result of Shvarts is generalized to an arbitrary compact Lie group action in the framework of the geometric approach and applied to estimating the genus of a free part of the unit sphere in the space of spherical harmonics under the natural representation of the group \(\text{SO}(n)\).